From bbb884ba6a54593ac1f103eef0fd8aa00a77c93d Mon Sep 17 00:00:00 2001 From: Floris van Doorn Date: Mon, 25 Mar 2024 19:17:00 +0100 Subject: [PATCH] add an actual test (only half) --- blueprint/src/bibliography.bib | 945 ++++++ blueprint/src/chapter/main.tex | 4435 ++++++++++++++++++++++++++++- blueprint/src/preamble/common.tex | 87 +- blueprint/src/preamble/print.tex | 8 - blueprint/src/preamble/web.tex | 9 - blueprint/src/print.tex | 14 +- blueprint/src/web.tex | 7 +- blueprint/tasks.py | 6 +- 8 files changed, 5453 insertions(+), 58 deletions(-) create mode 100644 blueprint/src/bibliography.bib diff --git a/blueprint/src/bibliography.bib b/blueprint/src/bibliography.bib new file mode 100644 index 00000000..ffecea8a --- /dev/null +++ b/blueprint/src/bibliography.bib @@ -0,0 +1,945 @@ +@incollection {MR3220096, + AUTHOR = {Assani, Idris and Presser, Kimberly}, + TITLE = {A survey of the return times theorem}, + BOOKTITLE = {Ergodic theory and dynamical systems}, + SERIES = {De Gruyter Proc. Math.}, + PAGES = {19--58}, + PUBLISHER = {De Gruyter, Berlin}, + YEAR = {2014}, + MRCLASS = {37A05 (37A30 37A45 37A50 37B20)}, + MRNUMBER = {3220096}, +} +@article {bl-generalized, + AUTHOR = {Bergelson, Vitaly and Leibman, Alexander}, + TITLE = {Distribution of values of bounded generalized polynomials}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {198}, + YEAR = {2007}, + NUMBER = {2}, + PAGES = {155--230}, + ISSN = {0001-5962}, + MRCLASS = {11K31 (11J54)}, + MRNUMBER = {2318563}, +MRREVIEWER = {Alexander Gorodnik}, + DOI = {10.1007/s11511-007-0015-y}, + URL = {https://doi.org/10.1007/s11511-007-0015-y}, +} + +@article {MR1037434, + AUTHOR = {Bourgain, J.}, + TITLE = {Double recurrence and almost sure convergence}, + JOURNAL = {J. Reine Angew. Math.}, + FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's + Journal]}, + VOLUME = {404}, + YEAR = {1990}, + PAGES = {140--161}, + ISSN = {0075-4102}, + MRCLASS = {28D05 (11K99 47A35)}, + MRNUMBER = {1037434}, +MRREVIEWER = {Anzelm Iwanik}, + DOI = {10.1515/crll.1990.404.140}, + URL = {https://doi.org/10.1515/crll.1990.404.140}, +} + + +@article {carleson, + AUTHOR = {Carleson, Lennart}, + TITLE = {On convergence and growth of partial sums of {F}ourier series}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {116}, + YEAR = {1966}, + PAGES = {135--157}, + ISSN = {0001-5962}, + MRCLASS = {42.11}, + MRNUMBER = {199631}, +MRREVIEWER = {J.-P. Kahane}, + DOI = {10.1007/BF02392815}, + URL = {https://doi.org/10.1007/BF02392815}, +} + +@book {MR0499948, + AUTHOR = {Coifman, Ronald R. and Weiss, Guido}, + TITLE = {Analyse harmonique non-commutative sur certains espaces + homog\`enes}, + SERIES = {Lecture Notes in Mathematics, Vol. 242}, + NOTE = {\'{E}tude de certaines int\'{e}grales singuli\`eres}, + PUBLISHER = {Springer-Verlag, Berlin-New York}, + YEAR = {1971}, + PAGES = {v+160}, + MRCLASS = {43A85 (22E30)}, + MRNUMBER = {0499948}, +} +@article {MR3021367, + AUTHOR = {Eisner, Tanja and Zorin-Kranich, Pavel}, + TITLE = {Uniformity in the {W}iener-{W}intner theorem for nilsequences}, + JOURNAL = {Discrete Contin. Dyn. Syst.}, + FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A}, + VOLUME = {33}, + YEAR = {2013}, + NUMBER = {8}, + PAGES = {3497--3516}, + ISSN = {1078-0947}, + MRCLASS = {37A45 (28D05 37A30)}, + MRNUMBER = {3021367}, +MRREVIEWER = {Nhan-Phu Chung}, + DOI = {10.3934/dcds.2013.33.3497}, + URL = {https://doi.org/10.3934/dcds.2013.33.3497}, +} + +@article {fefferman, + AUTHOR = {Fefferman, Charles}, + TITLE = {Pointwise convergence of {F}ourier series}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {98}, + YEAR = {1973}, + PAGES = {551--571}, + ISSN = {0003-486X}, + MRCLASS = {42A20}, + MRNUMBER = {340926}, +MRREVIEWER = {P. Sj\"{o}lin}, + DOI = {10.2307/1970917}, + URL = {https://doi.org/10.2307/1970917}, +} +@book {fisher-ruzhansky, + AUTHOR = {Fischer, Veronique and Ruzhansky, Michael}, + TITLE = {Quantization on nilpotent {L}ie groups}, + SERIES = {Progress in Mathematics}, + VOLUME = {314}, + PUBLISHER = {Birkh\"{a}user/Springer, [Cham]}, + YEAR = {2016}, + PAGES = {xiii+557}, + ISBN = {978-3-319-29557-2; 978-3-319-29558-9}, + MRCLASS = {22E25 (22E30 35R03 35S05 43A80 46L05)}, + MRNUMBER = {3469687}, +MRREVIEWER = {Antoni Wawrzy\'{n}czyk}, + DOI = {10.1007/978-3-319-29558-9}, + URL = {https://doi.org/10.1007/978-3-319-29558-9}, +} +@book {goodman, + AUTHOR = {Goodman, Roe W.}, + TITLE = {Nilpotent {L}ie groups: structure and applications to + analysis}, + SERIES = {Lecture Notes in Mathematics, Vol. 562}, + PUBLISHER = {Springer-Verlag, Berlin-New York}, + YEAR = {1976}, + PAGES = {x+210}, + MRCLASS = {22E25 (22E30 22E45 32M15 35H05)}, + MRNUMBER = {0442149}, +MRREVIEWER = {G. L. Litvinov}, +} +@article {MR2115460, + AUTHOR = {Grafakos, Loukas and Martell, Jos\'{e} Mar\'{\i}a and Soria, Fernando}, + TITLE = {Weighted norm inequalities for maximally modulated singular + integral operators}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {331}, + YEAR = {2005}, + NUMBER = {2}, + PAGES = {359--394}, + ISSN = {0025-5831}, + MRCLASS = {42B20 (47G10)}, + MRNUMBER = {2115460}, +MRREVIEWER = {Xavier Tolsa}, + DOI = {10.1007/s00208-004-0586-2}, + URL = {https://doi.org/10.1007/s00208-004-0586-2}, +} +@article {green-tao-quantitative, + AUTHOR = {Green, Ben and Tao, Terence}, + TITLE = {The quantitative behaviour of polynomial orbits on + nilmanifolds}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {175}, + YEAR = {2012}, + NUMBER = {2}, + PAGES = {465--540}, + ISSN = {0003-486X}, + MRCLASS = {37A15}, + MRNUMBER = {2877065}, +MRREVIEWER = {Tamar Ziegler}, + DOI = {10.4007/annals.2012.175.2.2}, + URL = {https://doi.org/10.4007/annals.2012.175.2.2}, +} +@article {Krause+2018, + AUTHOR = {Krause, Ben and Zorin-Kranich, Pavel}, + TITLE = {Weighted and vector-valued variational estimates for ergodic + averages}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {38}, + YEAR = {2018}, + NUMBER = {1}, + PAGES = {244--256}, + ISSN = {0143-3857}, + MRCLASS = {37A25 (60G46)}, + MRNUMBER = {3742545}, +MRREVIEWER = {Idris Assani}, + DOI = {10.1017/etds.2016.27}, + URL = {https://doi.org/10.1017/etds.2016.27}, +} +@article {MR2430729, + AUTHOR = {Lacey, Michael and Terwilleger, Erin}, + TITLE = {A {W}iener-{W}intner theorem for the {H}ilbert transform}, + JOURNAL = {Ark. Mat.}, + FJOURNAL = {Arkiv f\"{o}r Matematik}, + VOLUME = {46}, + YEAR = {2008}, + NUMBER = {2}, + PAGES = {315--336}, + ISSN = {0004-2080}, + MRCLASS = {42A20 (42A38)}, + MRNUMBER = {2430729}, + DOI = {10.1007/s11512-008-0080-2}, + URL = {https://doi.org/10.1007/s11512-008-0080-2}, +} + + + + + @article {lacey-thiele, + AUTHOR = {Lacey, Michael and Thiele, Christoph}, + TITLE = {A proof of boundedness of the {C}arleson operator}, + JOURNAL = {Math. Res. Lett.}, + FJOURNAL = {Mathematical Research Letters}, + VOLUME = {7}, + YEAR = {2000}, + NUMBER = {4}, + PAGES = {361--370}, + ISSN = {1073-2780}, + MRCLASS = {42A20 (42B10 42B25 47B38)}, + MRNUMBER = {1783613}, +MRREVIEWER = {Loukas Grafakos}, + DOI = {10.4310/MRL.2000.v7.n4.a1}, + URL = {https://doi.org/10.4310/MRL.2000.v7.n4.a1}, +} + +@article {leibman, + AUTHOR = {Leibman, A.}, + TITLE = {Polynomial mappings of groups}, + JOURNAL = {Israel J. Math.}, + FJOURNAL = {Israel Journal of Mathematics}, + VOLUME = {129}, + YEAR = {2002}, + PAGES = {29--60}, + ISSN = {0021-2172}, + MRCLASS = {20F18 (20K30 37A25 43A07 47D03)}, + MRNUMBER = {1910931}, +MRREVIEWER = {Tullio G. Ceccherini-Silberstein}, + DOI = {10.1007/BF02773152}, + URL = {https://doi.org/10.1007/BF02773152}, +} +@article {ramos, + AUTHOR = {Ramos, Jo\~{a}o P. G.}, + TITLE = {The {H}ilbert transform along the parabola, the polynomial + {C}arleson theorem and oscillatory singular integrals}, + JOURNAL = {Math. Ann.}, + FJOURNAL = {Mathematische Annalen}, + VOLUME = {379}, + YEAR = {2021}, + NUMBER = {1-2}, + PAGES = {159--185}, + ISSN = {0025-5831}, + MRCLASS = {42B20 (42B25 44A12)}, + MRNUMBER = {4211085}, +MRREVIEWER = {Joris Roos}, + DOI = {10.1007/s00208-020-02075-5}, + URL = {https://doi.org/10.1007/s00208-020-02075-5}, +} + + + + +@article{christ1990b, + title={A T(b) theorem with remarks on analytic capacity and the Cauchy integral}, + author={Christ, Michael}, + journal={Colloquium Mathematicum}, + volume={2}, + number={60-61}, + pages={601--628}, + year={1990} +} + +@book {hk-book, + AUTHOR = {Host, Bernard and Kra, Bryna}, + TITLE = {Nilpotent structures in ergodic theory}, + SERIES = {Mathematical Surveys and Monographs}, + VOLUME = {236}, + PUBLISHER = {American Mathematical Society, Providence, RI}, + YEAR = {2018}, + PAGES = {X+427}, + ISBN = {978-1-4704-4780-9}, + MRCLASS = {37-02 (11B30 28D05 37Axx 37B05 47A35)}, + MRNUMBER = {3839640}, +MRREVIEWER = {Yonatan Gutman}, + DOI = {10.1090/surv/236}, + URL = {https://doi.org/10.1090/surv/236}, +} +@article {ledonne-golo, + AUTHOR = {Le Donne, Enrico and Golo, Sebastiano Nicolussi}, + TITLE = {Metric {L}ie groups admitting dilations}, + JOURNAL = {Ark. Mat.}, + FJOURNAL = {Arkiv f\"{o}r Matematik}, + VOLUME = {59}, + YEAR = {2021}, + NUMBER = {1}, + PAGES = {125--163}, + ISSN = {0004-2080}, + MRCLASS = {54E40 (22E15 54E45)}, + MRNUMBER = {4256009}, +MRREVIEWER = {Xiaodan Zhou}, + DOI = {10.4310/arkiv.2021.v59.n1.a5}, + URL = {https://doi.org/10.4310/arkiv.2021.v59.n1.a5}, +} +@book {varadarajan, + AUTHOR = {Varadarajan, V. S.}, + TITLE = {Lie groups, {L}ie algebras, and their representations}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {102}, + NOTE = {Reprint of the 1974 edition}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1984}, + PAGES = {xiii+430}, + ISBN = {0-387-90969-9}, + MRCLASS = {22-01 (17-01)}, + MRNUMBER = {746308}, + DOI = {10.1007/978-1-4612-1126-6}, + URL = {https://doi.org/10.1007/978-1-4612-1126-6}, +} + +@book {knapp, + AUTHOR = {Knapp, Anthony W.}, + TITLE = {Lie groups beyond an introduction}, + SERIES = {Progress in Mathematics}, + VOLUME = {140}, + EDITION = {Second}, + PUBLISHER = {Birkh\"{a}user Boston, Inc., Boston, MA}, + YEAR = {2002}, + PAGES = {xviii+812}, + ISBN = {0-8176-4259-5}, + MRCLASS = {22-01}, + MRNUMBER = {1920389}, +} + +@article {donne-survey-carnot, + AUTHOR = {Le Donne, Enrico}, + TITLE = {A primer on {C}arnot groups: homogenous groups, + {C}arnot-{C}arath\'{e}odory spaces, and regularity of their + isometries}, + JOURNAL = {Anal. Geom. Metr. Spaces}, + FJOURNAL = {Analysis and Geometry in Metric Spaces}, + VOLUME = {5}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {116--137}, + MRCLASS = {53C17 (22E25 22F30 43A80)}, + MRNUMBER = {3742567}, +MRREVIEWER = {Andrea Pinamonti}, + DOI = {10.1515/agms-2017-0007}, + URL = {https://doi.org/10.1515/agms-2017-0007}, +} + +@article {lie-quadratic, + AUTHOR = {Lie, Victor}, + TITLE = {The (weak-{$L^2$}) boundedness of the quadratic {C}arleson + operator}, + JOURNAL = {Geom. Funct. Anal.}, + FJOURNAL = {Geometric and Functional Analysis}, + VOLUME = {19}, + YEAR = {2009}, + NUMBER = {2}, + PAGES = {457--497}, + ISSN = {1016-443X}, + MRCLASS = {42A50 (42A20)}, + MRNUMBER = {2545246}, +MRREVIEWER = {Javier Duoandikoetxea}, + DOI = {10.1007/s00039-009-0010-x}, + URL = {https://doi.org/10.1007/s00039-009-0010-x}, +} + +@article {lie-polynomial, + AUTHOR = {Lie, Victor}, + TITLE = {The polynomial {C}arleson operator}, + JOURNAL = {Ann. of Math. (2)}, + FJOURNAL = {Annals of Mathematics. Second Series}, + VOLUME = {192}, + YEAR = {2020}, + NUMBER = {1}, + PAGES = {47--163}, + ISSN = {0003-486X}, + MRCLASS = {42A20 (42A50)}, + MRNUMBER = {4125450}, +MRREVIEWER = {Michael T. Lacey}, + DOI = {10.4007/annals.2020.192.1.2}, + URL = {https://doi.org/10.4007/annals.2020.192.1.2}, +} + +@article {MR420837, + AUTHOR = {L\'{e}pingle, D.}, + TITLE = {La variation d'ordre {$p$} des semi-martingales}, + JOURNAL = {Z. Wahrscheinlichkeitstheorie und Verw. Gebiete}, + FJOURNAL = {Zeitschrift f\"{u}r Wahrscheinlichkeitstheorie und Verwandte + Gebiete}, + VOLUME = {36}, + YEAR = {1976}, + NUMBER = {4}, + PAGES = {295--316}, + MRCLASS = {60G45}, + MRNUMBER = {420837}, +MRREVIEWER = {Norihiko Kazamaki}, + DOI = {10.1007/BF00532696}, + URL = {https://doi.org/10.1007/BF00532696}, +} + +@article {nsw-balls, + AUTHOR = {Nagel, Alexander and Stein, Elias M. and Wainger, Stephen}, + TITLE = {Balls and metrics defined by vector fields. {I}. {B}asic + properties}, + JOURNAL = {Acta Math.}, + FJOURNAL = {Acta Mathematica}, + VOLUME = {155}, + YEAR = {1985}, + NUMBER = {1-2}, + PAGES = {103--147}, + ISSN = {0001-5962}, + MRCLASS = {46E35 (26D10 32F20 35H05 46N05 58G05)}, + MRNUMBER = {793239}, +MRREVIEWER = {Gerald B. Folland}, + DOI = {10.1007/BF02392539}, + URL = {https://doi.org/10.1007/BF02392539}, +} +@article {pansu-nil, + AUTHOR = {Pansu, Pierre}, + TITLE = {Croissance des boules et des g\'{e}od\'{e}siques ferm\'{e}es dans les + nilvari\'{e}t\'{e}s}, + JOURNAL = {Ergodic Theory Dynam. Systems}, + FJOURNAL = {Ergodic Theory and Dynamical Systems}, + VOLUME = {3}, + YEAR = {1983}, + NUMBER = {3}, + PAGES = {415--445}, + ISSN = {0143-3857}, + MRCLASS = {53C20 (20F18 22E25 58E10)}, + MRNUMBER = {741395}, +MRREVIEWER = {Victor Bangert}, + DOI = {10.1017/S0143385700002054}, + URL = {https://doi.org/10.1017/S0143385700002054}, +} +@article {tao-wright, + AUTHOR = {Tao, Terence and Wright, James}, + TITLE = {{$L^p$} improving bounds for averages along curves}, + JOURNAL = {J. Amer. Math. Soc.}, + FJOURNAL = {Journal of the American Mathematical Society}, + VOLUME = {16}, + YEAR = {2003}, + NUMBER = {3}, + PAGES = {605--638}, + ISSN = {0894-0347}, + MRCLASS = {42B15 (44A12 58D15)}, + MRNUMBER = {1969206}, +MRREVIEWER = {Josefina Alvarez}, + DOI = {10.1090/S0894-0347-03-00420-X}, + URL = {https://doi.org/10.1090/S0894-0347-03-00420-X}, +} + +@book {corwin-greenleaf, + AUTHOR = {Corwin, Lawrence J. and Greenleaf, Frederick P.}, + TITLE = {Representations of nilpotent {L}ie groups and their + applications. {P}art {I}}, + SERIES = {Cambridge Studies in Advanced Mathematics}, + VOLUME = {18}, + NOTE = {Basic theory and examples}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {1990}, + PAGES = {viii+269}, + ISBN = {0-521-36034-X}, + MRCLASS = {22E27 (22-01 22E25 22E30)}, + MRNUMBER = {1070979}, +MRREVIEWER = {Jeffrey Fox}, +} + +@article {zk-polynomial, + AUTHOR = {Zorin-Kranich, Pavel}, + TITLE = {Maximal polynomial modulations of singular integrals}, + JOURNAL = {Adv. Math.}, + FJOURNAL = {Advances in Mathematics}, + VOLUME = {386}, + YEAR = {2021}, + PAGES = {Paper No. 107832, 40}, + ISSN = {0001-8708}, + MRCLASS = {42B20 (42B25)}, + MRNUMBER = {4270523}, +MRREVIEWER = {Marco Bramanti}, + DOI = {10.1016/j.aim.2021.107832}, + URL = {https://doi.org/10.1016/j.aim.2021.107832}, +} + +@misc{zk-var-trunc, + doi = {10.48550/ARXIV.2009.04541}, + + url = {https://arxiv.org/abs/2009.04541}, + + author = {Zorin-Kranich, Pavel}, + + keywords = {Classical Analysis and ODEs (math.CA), FOS: Mathematics, FOS: Mathematics, 42B20 (Primary) 42B25 (Secondary)}, + + title = {Variational truncations of singular integrals on spaces of homogeneous type}, + + publisher = {arXiv}, + + year = {2020}, + + copyright = {Creative Commons Attribution 4.0 International}, +} + + +@article {mnatsakanyan, + AUTHOR = {Mnatsakanyan, Gevorg}, + TITLE = {On almost-everywhere convergence of {M}almquist-{T}akenaka + series}, + JOURNAL = {J. Funct. Anal.}, + FJOURNAL = {Journal of Functional Analysis}, + VOLUME = {282}, + YEAR = {2022}, + NUMBER = {12}, + PAGES = {Paper No. 109461, 33}, + ISSN = {0022-1236}, + MRCLASS = {42A20 (42A50)}, + MRNUMBER = {4395336}, +MRREVIEWER = {Elijah Liflyand}, + DOI = {10.1016/j.jfa.2022.109461}, + URL = {https://doi.org/10.1016/j.jfa.2022.109461}, +} + +@book {bonfiglioli, + AUTHOR = {Bonfiglioli, A. and Lanconelli, E. and Uguzzoni, F.}, + TITLE = {Stratified {L}ie groups and potential theory for their + sub-{L}aplacians}, + SERIES = {Springer Monographs in Mathematics}, + PUBLISHER = {Springer, Berlin}, + YEAR = {2007}, + PAGES = {xxvi+800}, + ISBN = {978-3-540-71896-3; 3-540-71896-6}, + MRCLASS = {22E30 (31C45 35-02 35H10 43A80)}, + MRNUMBER = {2363343}, +MRREVIEWER = {Maria Stella Fanciullo}, +} + + +@book {stein-book, + AUTHOR = {Stein, Elias M.}, + TITLE = {Harmonic analysis: real-variable methods, orthogonality, and + oscillatory integrals}, + SERIES = {Princeton Mathematical Series}, + VOLUME = {43}, + NOTE = {With the assistance of Timothy S. Murphy, + Monographs in Harmonic Analysis, III}, + PUBLISHER = {Princeton University Press, Princeton, NJ}, + YEAR = {1993}, + PAGES = {xiv+695}, + ISBN = {0-691-03216-5}, + MRCLASS = {42-02 (35Sxx 43-02 47G30)}, + MRNUMBER = {1232192}, +MRREVIEWER = {Michael Cowling}, +} +@article {Hyt+adjacent, + AUTHOR = {Hyt\"{o}nen, Tuomas and Kairema, Anna}, + TITLE = {Systems of dyadic cubes in a doubling metric space}, + JOURNAL = {Colloq. Math.}, + FJOURNAL = {Colloquium Mathematicum}, + VOLUME = {126}, + YEAR = {2012}, + NUMBER = {1}, + PAGES = {1--33}, + ISSN = {0010-1354}, + MRCLASS = {42B25 (60D05)}, + MRNUMBER = {2901199}, +MRREVIEWER = {Raymond H. Cox}, + DOI = {10.4064/cm126-1-1}, + URL = {https://doi.org/10.4064/cm126-1-1}, +} +@article {PSquasimetric, + AUTHOR = {Paluszy\'{n}ski, Maciej and Stempak, Krzysztof}, + TITLE = {On quasi-metric and metric spaces}, + JOURNAL = {Proc. Amer. Math. Soc.}, + FJOURNAL = {Proceedings of the American Mathematical Society}, + VOLUME = {137}, + YEAR = {2009}, + NUMBER = {12}, + PAGES = {4307--4312}, + ISSN = {0002-9939}, + MRCLASS = {54E35}, + MRNUMBER = {2538591}, + DOI = {10.1090/S0002-9939-09-10058-8}, + URL = {https://doi.org/10.1090/S0002-9939-09-10058-8}, +} + +@article {stein-wainger, + AUTHOR = {Stein, Elias M. and Wainger, Stephen}, + TITLE = {Oscillatory integrals related to {C}arleson's theorem}, + JOURNAL = {Math. Res. Lett.}, + FJOURNAL = {Mathematical Research Letters}, + VOLUME = {8}, + YEAR = {2001}, + NUMBER = {5-6}, + PAGES = {789--800}, + ISSN = {1073-2780}, + MRCLASS = {42B20 (47G10)}, + MRNUMBER = {1879821}, +MRREVIEWER = {B. S. Rubin}, + DOI = {10.4310/MRL.2001.v8.n6.a9}, + URL = {https://doi.org/10.4310/MRL.2001.v8.n6.a9}, +} + +@article {oberlinetal, + AUTHOR = {Oberlin, Richard and Seeger, Andreas and Tao, Terence and + Thiele, Christoph and Wright, James}, + TITLE = {A variation norm {C}arleson theorem}, + JOURNAL = {J. Eur. Math. Soc. (JEMS)}, + FJOURNAL = {Journal of the European Mathematical Society (JEMS)}, + VOLUME = {14}, + YEAR = {2012}, + NUMBER = {2}, + PAGES = {421--464}, + ISSN = {1435-9855}, + MRCLASS = {42A20 (42A16 42A45 42A61)}, + MRNUMBER = {2881301}, +MRREVIEWER = {Alexander V. Tovstolis}, + DOI = {10.4171/JEMS/307}, + URL = {https://doi.org/10.4171/JEMS/307}, +} + +@article {lacey-terwilleger, + AUTHOR = {Lacey, Michael and Terwilleger, Erin}, + TITLE = {A {W}iener-{W}intner theorem for the {H}ilbert transform}, + JOURNAL = {Ark. Mat.}, + FJOURNAL = {Arkiv f\"{o}r Matematik}, + VOLUME = {46}, + YEAR = {2008}, + NUMBER = {2}, + PAGES = {315--336}, + ISSN = {0004-2080}, + MRCLASS = {42A20 (42A38)}, + MRNUMBER = {2430729}, + DOI = {10.1007/s11512-008-0080-2}, + URL = {https://doi.org/10.1007/s11512-008-0080-2}, +} + +@book {assani-book, + AUTHOR = {Assani, Idris}, + TITLE = {Wiener {W}intner ergodic theorems}, + PUBLISHER = {World Scientific Publishing Co., Inc., River Edge, NJ}, + YEAR = {2003}, + PAGES = {xii+216}, + ISBN = {981-02-4439-8}, + MRCLASS = {37A30 (28D05 47A35 82B03)}, + MRNUMBER = {1995517}, +MRREVIEWER = {U. Krengel}, + DOI = {10.1142/4538}, + URL = {https://doi.org/10.1142/4538}, +} +@article {assani-survey, + AUTHOR = {Assani, Idris}, + TITLE = {Wiener-{W}intner ergodic theorem, in brief}, + JOURNAL = {Notices Amer. Math. 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Math.}, + FJOURNAL = {Inventiones Mathematicae}, + VOLUME = {231}, + YEAR = {2023}, + NUMBER = {3}, + PAGES = {1023--1140}, + ISSN = {0020-9910}, + MRCLASS = {37A30 (11B30 22E25)}, + MRNUMBER = {4549088}, + DOI = {10.1007/s00222-022-01159-0}, + URL = {https://doi.org/10.1007/s00222-022-01159-0}, +}, + +@article{hebisch1990smooth, + title={A smooth subadditive homogeneous norm on a homogeneous group}, + author={Hebisch, Waldemar and Sikora, Adam}, + journal={Studia Mathematica}, + volume={96}, + pages={231--236}, + year={1990}, + publisher={Instytut Matematyczny Polskiej Akademii Nauk} +}, + +@article{nicolaides1972class, + title={On a class of finite elements generated by Lagrange interpolation}, + author={Nicolaides, RA}, + journal={SIAM Journal on Numerical Analysis}, + volume={9}, + number={3}, + pages={435--445}, + year={1972}, + publisher={SIAM} +} \ No newline at end of file diff --git a/blueprint/src/chapter/main.tex b/blueprint/src/chapter/main.tex index 2e9868e0..daa57a66 100644 --- a/blueprint/src/chapter/main.tex +++ b/blueprint/src/chapter/main.tex @@ -1,5 +1,4438 @@ % This is the main point of entry to the blueprint. % Add chapters of the blueprint here. % This file is not meant to be built. Build src/web.tex or src/print.text instead. +\title{Carleson operators on doubling metric measure spaces} -\input{chapter/jensen} +\author{Lars Becker \and Floris van Doorn \and Asgar Jamneshan \and Rajula Srivastava \and Christoph Thiele} + +\date{\today} + +\begin{abstract} +We prove bounds for a generalization of Carleson operators on doubling metric measure spaces. +This paper is written as a blueprint for a computer verified proof. We also use our theorem to give a blueprint for a computer verification of Carleson's classical theorem on almost everywhere convergence of Fourier series. +\end{abstract} + +\maketitle + +\tableofcontents + +\chapter{Introduction} + +In his \cite{carleson} paper, Lennart Carleson +answered a classical question on convergence of +Fourier series of continuous functions. Theorem +\ref{classical} is a version of his result. + +Let $f$ be a complex valued, $2\pi $-periodic bounded Borel measurable function on the real line and given an integer $n$, define the Fourier coefficient +\begin{equation} + \widehat{f}_n:=\frac {1}{2\pi} \int_0^{2\pi} f(x) e^{- i nx} dx . +\end{equation} +Define for $N\ge 0$ the partial Fourier sum +\begin{equation} + S_Nf(x):=\sum_{n=-N}^N \widehat{f}_n e^{i nx}\ . +\end{equation} +\begin{theorem}\label{classical} +\uses{thm main 1, thm main 1} +Let $f$ be a $2\pi $-periodic complex valued uniformly continuous function an $\R$ that satisfies the bound +$|f(x)|\le 1$ for all $x\in \R$. For all $0<\epsilon\le 2\pi$, +there exists a Borel set $E\subset [0,2\pi]$ with measure +$|E|\le \epsilon$ and a positive integer $N_0$ such that for all +$\theta\in [2\pi]\setminus E$ \rs{$[0, 2\pi]\setminus E$} and all integers $N>N_0$ we have +\begin{equation}\label{aeconv} +|f(x)-S_N f(x)|\le \epsilon. +\end{equation} +\end{theorem} +\rs{Is $\theta$ supposed to be $x$?} + +Note that mere continuity implies uniform continuity +in the setting of this theorem. Applying this theorem +with a sequence of $\epsilon_n:= 2^{-n}\delta$ for $n\ge 1$ +and taking the union of corresponding exceptional sets $E_n$, we see that +outside a set of measure $\delta$, the partial Fourier sums +converges pointwise for $N\to \infty$. Applying this with a sequence +of $\delta$ shrinking to zero and taking the intersection of the corresponding exceptional +sets, which has measure zero, we see that the Fourier series converges outside +a set of measure zero. This is another classical formulation of the theorem of Carleson. + +This paper provides a blue-print for a computer verification of Theorem \ref{classical}, which we plan to code in Lean. +We pass through a novel bound for a +generalization of the so-called Carlson operator +to doubling metric measure spaces \rs{Should also mention polynomials/general class of functions}. This generalization is of its own interest as new result and we state it as our main novel theorem in this paper. + +The maximally truncated generalized Carleson operator $T$, which we simply refer to as Carleson operator, shall be defined by + \begin{equation} + \label{def main op} + Tf(x):=\sup_{\mfa\in\Mf} \sup_{0 < R_1 < R_2}\left| \int_{R_1 < \rho(x,y) < R_2} K(x,y) f(y) e(\mfa(y)) \, \mathrm{d}\mu(y) \right|\, , +\end{equation} +where $e(r)=e^{ir}$ for any real number $r$. +For example, in +\cite{zk-polynomial}, the underlying space for the variables $x,y$ is the Euclidean space $\R^{\bf d}$, $\rho$ is the Euclidean metric, $\mu$ is the Lebesgue measure, +$K$ is a Calder\'on--Zygmund kernel with some H\"older regularity, +and $\Mf$ is the +class of polynomials up to some degree $d$. + +%Estimates of interest for the Carleson operator include bounds from $L^q(\R^{\bf d})$ to itself for $10$ we have +\begin{equation}\label{doublingx} + \mu(B(x,2R))\le 2^a\mu(B(x,R))\,, +\end{equation} +where we have denoted by $B(x,R)$ the open ball of radius $R$ centred at $x$: +\begin{equation}\label{eq define ball} + B(x,R):=\{y\in X: \rho(x,y)2$ and +${\bf{d}=1}$ is the polynomial Carleson operator and estimated in \cite{lie-polynomial}. The case of the class of polynomials with +vanishing linear coefficient is simpler and was estimated in \cite{stein-wainger}. +Adapting the proof in \cite{zk-polynomial} through +a number of axiomatic properties of $\Mf$ was already done +in \cite{mnatsakanyan} to pass to classes of Blaschke factors on the disc rather than polynomials. An application of the quadratic Carleson +operator of \cite{lie-quadratic} appears in \cite{ramos}. \rs{The variables $d$ and ${\bf d}$ are currently undefined} + +\noindent \textit{Acknowledgement.} +L.B., R.S., and C.T. were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -- EXC-2047/1 -- 390685813 as well as SFB 1060. +A.J. is funded by the T\"UBITAK (Scientific and Technological Research Council of T\"urkiye) under the 1001-project 123F122. + +\chapter{Overview of the proof of Theorem \ref{thm main 1}} +\label{overviewsection} + + +This section organizes the proof of Theorem +\ref{thm main 1} into the subsequent five +mutually independent sections. +It formulates four auxiliary propositions, each proved + in one sections. The fifth section proves Theorem \ref{thm main 1}. +The present section also introduces all definitions and statements used across boundaries of these five sections. + + +Let $a, q, \tau$ be given as in Theorem \ref{thm main 1} +and set $A:=2^a$. +For two complex quantities $X,Y$, usually depending on +$A, q, \tau$ and some further parameters, +we write $X\lesssim Y$ if there exists + \[C(a,\tau,q)>0\] +such that for all values of the further parameters +\[|X|\le C(a,\tau,q)|Y|.\] Note that $\lesssim$ is transitive. + + + +Define +\begin{equation}\label{defineD} +D:= 2^{100 a^2}\, . +\end{equation} +Define +\begin{equation}\label{definekappa} +\kappa:= \, . +\end{equation} +\ct{fill in after completing grid section}. + +Let + $\psi:\R \to \R$ be the unique compactly supported, piece-wise affine, linear, continuous function with corners precisely at $\frac 1{4D}$, $\frac 1{2D}$, $\frac 14$ and $\frac 12$ which satisfies + \begin{equation} + \label{eq psisum} + \sum_{s\in \mathbb{Z}} \psi(D^{-s}x)=1 . +\end{equation} +for all $x>0$. This function vanishes outside $[\frac1{4D},\frac 12]$, is constant one on +$[\frac1{2D},\frac 14]$, and is Lipschitz +with constant $4D$. + + + + + + + +Let a doubling metric measure space $(X,\rho,\mu, A)$ be given. +Let a $\tau$-cancellative compatible collection $\Mf$ of functions on $X$ be given. \rs{This is not consistent with the notation in Section 1, which would instead have us write $(X,\rho,\mu, \log _2 A)$} +Let $o\in X$ be a point such that $\mfa(o)=1$ +for all $\mfa\in \Mf$. + +\begin{lemma} + For any ball $B$ in $X$, the local oscillation +$d_{B}$ is a metric on $\Mf$. +\end{lemma} + +\begin{proof} + Symmetry and triangle inequality are immediate, and we argue that +for any ball $B(x,r)$, the identity +\begin{equation}\label{dvanish} +d_{B(x,r)}(\mfa,\mfb)=0 +\end{equation} +implies $\mfa(y)=\mfb(y)$ for all $y\in X$. Assume $\eqref{dvanish}$. +Let +\begin{equation} + R=1+\rho(x,o)+\rho(x,y)\, . +\end{equation} +By an iterated application of +the comparability of the norms \eqref{firstdb} or \eqref{seconddb} \rs{No relation between $r$ and $R$ right?} +\begin{equation} + d_{B(x,R)}(\mfa,\mfb)=0. +\end{equation} +By the definition of +the local oscillation, evaluated using $o$ as one of the points and $y$ as the other, we obtain +$\mfa(y)=\mfb(y)$. +\end{proof} +%Let $\mathcal{B}(\Mf)$ denote the Borel $\sigma$-algebra in $\Mf$ with respect to the +%unique topology generated by any of the metrics $d_{B}$ with respect to some non-empty ball $B$ in $X$. \ct{Not sure we need that (?) move to section 3} + +Let a one-sided $\tau$-Calder\'on--Zygmund kernel $K$ on $X$ be given so that the operator $T_*$ defined in \eqref{def tang unm op} +satisfies +\eqref{nontanbound}. Let $T$ be the corresponding operator as defined in \eqref{def main op}. + + +For $s\in\mathbb{Z}$, we define +\begin{equation}\label{defks} + K_s(x,y):=K(x,y)\psi(D^{-s}\rho(x,y))\,, +\end{equation} +so that for each $x, y \in X$ with $x\neq y$ we have +$$K(x,y)=\sum_{s\in\mathbb{Z}}K_s(x,y).$$ + In Section \ref{thmfromproplinear}, we prove Theorem \ref{thm main 1} + from the more finitary version, Proposition \ref{prop-linear} below. We call a function from a measure space to a finite set measurable if the pre-image of each of the elements in the range is measurable. + \lars{Use consistent notation $\mathbf{1}$ or $1$ for indicator functions} + +\begin{prop}\label{prop-linear} +Let ${\sigma_1},\sigma_2\colon X\to \mathbb{Z}$ be measurable functions with finite range and ${\sigma_1}\leq \sigma_2$. Let $\tQ\colon X\to \Mf$ be a measurable function with finite range. Let $F,G$ be bounded Borel sets in $X$. Then there is a Borel set $G'$ in $X$ with $\mu(G')\leq \frac 12 \mu(G)$ such that +for all Borel functions $f:X\to \C$ with $|f|\le \mathbf{1}_F$. +\begin{multline}\label{eq-linearized} +\int_{G \setminus G'} \left|\sum_{s={\sigma_1}(x)}^{{\sigma_2}(x)} \int K_s(x,y) f(y) \tQ(x)(y) \, \mathrm{d}\mu(y) \right| \mathrm{d}\mu(x) \lesssim \mu(G)^{\frac 1{q'}} + \mu(F)^{\frac 1 q}\,. +\end{multline} +\end{prop} +Let measurable functions ${\sigma_1}\leq \sigma_2\colon X\to \mathbb{Z}$ with finite range be given. let a measurable function +$\tQ\colon X\to \Mf$ with finite range +be given. +Let bounded Borel sets $F,G$ in $X$ be given. +Let $S$ be the smallest integer such that the ranges of +$\underline{\sigma}$ and $ \overline\sigma$ are contained in $[-S,S]$ and $F$ and $G$ are contained +in the ball $B(o, D^S)$. + + +In Section \ref{christsection}, +we prove Proposition \ref{prop-linear} + using a +bound for a dyadic model formulated in Proposition +\ref{prop dyadic} below. + + +A grid structure $(\mathcal{D}, c, s)$ on $X$ consists of a finite collection $\mathcal{D}$ of Borel sets in $X$ called dyadic cubes, a surjective function $s\colon \mathcal{D}\to [-S, S]$ +called scale function, +and a function $c:\mathcal{D}\to X$ +called center function such that the five properties +\eqref{coverdyadic}, +\eqref{dyadicproperty}, \eqref{coverball}, +\eqref{eq vol sp cube}, and \eqref{eq small boundary} +hold. + +For each dyadic cube $I$ and each $-S\le k0$, +\begin{equation} + \label{eq small boundary} + \mu(\{x \in I \ : \ \rho(x, X \setminus I) \leq t D^{s(I)}\}) \le 2^{2a+2} t^\kappa \mu(I)\,. + \end{equation} +\ct{probably there is a better way to formulate this, following +Lemma 4.10 or so} + + + + + + + + +A tile structure $(\fP,\sc,\fc,\fcc,\pc,\ps)$ +for a given grid structure $(\mathcal{D}, c, s)$ +is a finite set $\fP$ of elements called tiles with five maps +\begin{align*} +\sc&\colon \fP\to {\mathcal{D}}\\ +\fc&\colon \fP\to \mathcal{P}(\Mf) \\ +\fcc &\colon \fP\to \Mf\\ +\pc &\colon \fP\to X\\ +\ps &\colon \fP\to \mathbb{Z} +\end{align*} +with $\sc$ surjective and $\mathcal{P}(\Mf)$ denoting the power set of $\Mf$ such that the five Properties \eqref{eq dis freq cover}, \eqref{eq dis freq cover}, +\eqref{eq freq comp ball}, \ref{tilecenter}, and +\ref{tilescale} hold. + +For each dyadic cube $I$, the restriction of the map $\Omega$ to the set +\begin{equation}\label{injective} + \fP(I)=\{\fp: \sc(\fp) =I\} +\end{equation} +is injective +and we have the disjoint covering property +\begin{equation}\label{eq dis freq cover} +\tQ(X)\subset \dot{\bigcup}_{\fp\in \fP(I)}\fc(\fp). +\end{equation} +For any tiles $\fp,\fq$ with $\sc(\fp)\subset \sc(\fq)$ and $\fc(\fp) \cap \fc(\fq) \neq \emptyset$ we have +\begin{equation} \label{eq freq dyadic} +\fc(\fq)\subset \fc(\fp) . +\end{equation} +For each tile $\fp$, + \begin{equation}\label{eq freq comp ball} + \fcc(\fp)\in B_{\fp}(\fcc(\fp), 0.2) \subset \fc(\fp) \subset B_{\fp}(\fcc(\fp),1)\,, + \end{equation} + where +\begin{equation} + B_{\fp} := \{\mfb \in \Mf \, : \, d_{\fp}(\mfa, \mfb) < R\,\} , +\end{equation} + and +\begin{equation}\label{defdp} +d_{\fp} := d_{B(\pc(\fp),\frac 14 D^\ps(\fp))}\, . +\end{equation} + + + + +We have for each tile $\fp$ +\begin{equation}\label{tilecenter} + \pc(\fp)=c(\sc(\fp)), +\end{equation} +\begin{equation}\label{tilescale} + \ps(\fp)=s(\sc(\fp)). +\end{equation} + + +\begin{prop} +\label{prop dyadic} +Let $(\mathcal{D}, c, s)$ be a grid structure and $(\fP,\sc,\fc,\fcc,\pc,\ps)$ +a tile structure for this grid structure. + +Define for $\fp\in \fP$ +\begin{equation}\label{defineep} + E(\fp)=\{x\in \sc(\fp): \tQ(x)\in \fc(\fp) , {\sigma_1}(x)\le \ps(\fp)\le {\sigma_2}(x)\} +\end{equation} +and +\begin{equation}\label{definetp} + T_{\fp} f(x)= 1_{E(\fp)}(x) \int K_{\ps(\fp)}(x,y) f(y) \tQ(x)(y)\overline{\tQ(x)(x)}\, d\mu(y). +\end{equation} +Then there exists a Borel set $G'\subset G$ with $\mu(G') \leq 1/2\mu(G)$ such that for all $f:X\to \C$ with $|f|\le \mathbf{1}_F$and all +$g:X\to \C$ with $|g|\le \mathbf{1}_{G\setminus G'}$ +we have +\begin{equation} + \label{disclesssim} + \int g(x) \sum_{\fp \in \fP} T_{\fp} f (x) \, \mathrm{d}\mu(x) \lesssim \mu(G)^{1/q'} \mu(F)^{1/q}\,. +\end{equation} +\end{prop} + +\lars{Why is there a dualizing function $g$ in the above Proposition? I don't think it is needed for the restricted weak type bounds, and it is not there for example in Proposition 2.2} + + + + + + + +The proof of Proposition \ref{prop dyadic} is done in Section \ref{proptopropprop} +by a reduction to two further propositions that we state below. + + +Fix a grid structure $(\mathcal{D}, c, s)$ and a tile structure $(\fP,\sc,\fc,\fcc,\pc,\ps)$ +for this grid structure. + +We define the relation +\begin{equation}\label{straightorder} + \fp\le \fp' +\end{equation} + on $\fP\times \fP$ meaning +$\sc(\fp)\subset \sc(\fp')$ and +$\Omega(\fp')\subset \Omega(\fp)$. +We further define for $\lambda,\kappa>0$ +the relation +\begin{equation}\label{wiggleorder} + \lambda \fp \lesssim \kappa\fp' +\end{equation} +on $\fP\times \fP$ meaning +$\sc(\fp)\subset \sc(\fp')$ and +\begin{equation} + B_{\fp'}(Q(\fp'),\kappa ) + \subset B_{\fp}(Q(\fp),\lambda )\, . +\end{equation} + + + +Define for a tile $\fp$ and $\lambda>0$ +\begin{equation}\label{definee1} + E_1(\fp):=\{x\in \sc(\fp)\cap G: \tQ(x)\in \fc(\fp)\}\, , +\end{equation} +\begin{equation}\label{definee2} + E_2(\lambda, \fp):=\{x\in \sc(\fp)\cap G: \tQ(x)\in B_{\fp}(\fcc(\fp), \lambda)\}\, . +\end{equation} + + + +Given a subset $\fP'$ of $\fP$, we define +$\fP(\fP')$ to be the set of +all $\fp \in \fP$ such that there exist $\fp' \in \fP'$ with $\sc(\fp)\subset \sc(\fp')$. Define the densities +\begin{equation}\label{definedens1} + {\dens}_1(\fP') := \sup_{\fp'\in \fP'}\sup_{\lambda \geq 2} \lambda^{-a} \sup_{\fp \in \fP(\fP'), \lambda \fp' \lesssim \lambda \fp} + \frac{\mu({E}_2(\lambda, \fp))}{\mu(\sc(\fp))}\, , +\end{equation} +\begin{equation}\label{definedens2} + {\dens}_2(\fP') := \sup_{\fp'\in \fP'} + \sup_{r>4D^{\ps(\fp)}} + \frac{\mu(F\cap B(\pc(\fp),r))}{\mu(B(\pc(\fp),r))}\, . +\end{equation} + + + + + +An antichain is a subset $\mathfrak{A}$ +of $\fP$ such that for any distinct $\fp,\fq\in \mathfrak{A}$ we do not have have $\fp\le \fq$. + +The following proposition is proved in Section \ref{antichainboundary}. +We choose a sufficiently small $\delta > 0$ depending on $A, \tau$, the precise choice will be made in Lemma \ref{SeparatedTrees}. +Depending on $\kappa$, $A$, $\tau$, $q$, we choose a sufficiently small $\epsilon > 0$, the exact choice will be implicit in the proofs of Proposition \ref{antichainprop} and Proposition \ref{forestprop}. \ct{This will be modified and psosibly moved around, +along with the definition of $\kappa$, when these sections are done.} + + +\begin{prop}\label{antichainprop} +For any antichain $\mathfrak{A} $ and for all $f:X\to \C$ with $|f|\le \mathbf{1}_F$ and all +$g:X\to \C$ with $|g|\le \mathbf{1}_{G}$ +\begin{equation} \label{eq antiprop} + |\int \overline{g(x)} \sum_{\fp \in \mathfrak{A}} T_{\fp} f(x)\, d\mu(x)| + \end{equation} + \begin{equation} + \le 2^{200a^3}({q}-1)^{-1} \tau^{-1}\dens_1(\mathfrak{A})^{\frac {(q-1)\tau^2}{8a^2}}\dens_2(\mathfrak{A})^{\frac 1{q}-\frac 12} \|f\|_2\|g\|_2\, . + \end{equation} +\end{prop} + +Let $n\ge 0$. +An $n$-forest is a pair $(\fU, \mathfrak{T})$ +where $\fU$ is a subset of $\fP$ +and $\mathfrak{T}$ is a map assigning to +each $\fu\in \fU$ a set $\fT (\fu)\subset \fP$ called tree +such that the following properties +\eqref{forest1}, \eqref{forest2}, +\eqref{forest3}, +\eqref{forest4}, +\eqref{forest5}, and +\eqref{forest6} +hold. + +For each $\fu\in \fU$ and each $\fp\in \fT(\fu)$ +we have +\begin{equation}\label{forest1} +4\fp\lesssim 1\fu. +\end{equation} + +For each $\fu, \in \fU$ and each $\fp,\fp''\in \fT(\fu)$ and $\fp'\in \fP$ +we have +\begin{equation}\label{forest2} + \fp, \fp'' \in \mathfrak{T}(\fu), \fp \leq \fp' \leq \fp'' \implies \fp' \in \mathfrak{T}(\fu). +\end{equation} +We have +\begin{equation}\label{forest3} + \|\sum_{\fu\in \fU} \mathbf{1}_{\sc(\fu))}\|_\infty \leq 2^n\,. +\end{equation} +We have for every $\fu\in \fU$ +\begin{equation}\label{forest4} +\dens_1(\fT(\fu))\le 2^{4a + 1-n}\, . +\end{equation} +With $Z=3/\delta$ we have for $\fu, \fu'\in \fU$ and $\fp\in \fT(\fu')$ with $\sc(\fp)\subset \sc(\fu)$ that +\begin{equation}\label{forest5} +d_{\fp}(\fcc(\fp), \fcc(\fu))>2^{Z(n+1)}\, . +\end{equation} +We have for every $\fu\in \fU$ and $\fp\in \fT(\fu)$ that +\begin{equation}\label{forest6} +B(\pc(\fp)), 8D^{\ps(\fp)})\subset \sc(\fu)). +\end{equation} + +%For $\fp\in \fP$ and $Q\in\mathcal{Q}$, we define their {separation} to be +%\[\Delta(\fp, Q):=d_{\sc(\fp)}(Q(\fp), Q)+1.\] + %A pair of boundary parts $\mathfrak{B}_1, \mathfrak{B}_2$ is \emph{$\Delta$-separated} if +% for $i \neq j$, $\fp \in \mathfrak{B}_i$ and $I(\fp) \subset I(\tp(\mathfrak{B}_j))$ implies $\Delta(\fp, Q({\tp(\mathfrak{B}_j)})) > \Delta$. + + +%A boundary part is a tree $\mathfrak{B}$, such that +%$$ +% B(x_{\sc(\fp)}, 8A^3 D^{s(\sc(\fp))}) \not\subset \sc(\tp(\mathfrak{B})) +%$$ +%for all tiles $\fp \in \mathfrak{B}$, and an $(n,k)$-boundary forest is a forest consisting only of boundary parts. + +%The following proposition is also proved in Section \ref{antichainboundary}. +%\begin{prop}\label{boundaryprop} +% For any $n \geq 1$ and any $(n,k)$-boundary forest $\mathfrak{F} \subset \fP_t$ +% we have +% $$\|\sum_{\fp\in \mathfrak{F}} T_{\fp} 1_F\|_{2\to 2} +% \lesssim 2^{-\epsilon n} \log(2 + k) t^{1/q - 1/2}.$$ +%\end{prop} + + + + +The following proposition is proved in Section \ref{treesection}. +\begin{prop}\label{forestprop} +For any $n\ge 1$ and any $n$-forest $(\fU,\fT)$ we have +$$\|\sum_{\fu\in \fU} +\sum_{\fp\in \fT(\fu)} T_{\fp} 1_F\|_{2\to 2} +\lesssim 2^{-\epsilon n} +\dens_2(\bigcup_{\fu\in \fU}\fT(\fu))^{1/q-1/2} \, .$$ +\end{prop} + +Theorem \ref{thm main 1} is formulated at the level of genereality +for general kernels satisfying the mere H\"older regularity condition \eqref{eqkernel y smooth}. On the other hand, the $\tau$-cancellative condition \eqref{ew vdc cond} is a testing against more regular, +namely Lipschitz functions. To bridge the gap, we follow \cite{zk-polynomial} to observe a variant of \eqref{ew vdc cond} that we formulate +in the following Proposition proved in Section \ref{liphoel}. + + +Define for any open ball $B$ of radius $R$ in $X$ the $\tau$-H\"older norm by +$$ + \|\varphi\|_{C^\tau(B)} = \sup_{x \in B} |\varphi(x)| + R^\tau \sup_{x,y \in X, x \neq y} \frac{|\varphi(x) - \varphi(y)|}{\rho(x,y)^\tau}\,. +$$ +\begin{prop} + \label{lem vdc regularity} + Let $z\in X$ and $R>0$ and set $B=B(z,R)$. + Let $\varphi: X \to \mathbb{C}$ by + supported on $B$ and satisfy $\|{\varphi}\|_{C^\tau(B)}<\infty$. + Let $\mfa, \mfb \in \mathcal{Q}$. Then + \begin{equation} + \label{eq vdc cond tau 2} + |\int e(\mfa(x)-{\mfb(x)})\varphi(x) dx|\le + 2^{4a} \mu(B) \|{\varphi}\|_{C^\tau(B)} + (1 + d_{B}(\mfa,\mfb))^{-\tau^2/(2+a)} + \,. + \end{equation} + \end{prop} + +\lars{Is this covering Lemma not usually called Vitali's covering Lemma? Besciovitch is a different one.} +We further formulate a classical Besicovitch covering result +and maximal function estimate that we need throughout several sections. +This following proposition will typically be applied to the absolute value of a complex valued function and be proved in Section \ref{sec hlm}. By a ball $B'$ we mean a set $B(x,r)$ with $x\in X$ +and $r>0$ as defined in \eqref{eq define ball}. +For a finite collection $\mathcal{B}$ of balls in $X$ +and $1\le q< \infty$ \rs{$q$ has already been fixed at the beginning of this section} define the measurable function $M_{\mathcal{B},q}h$ on $X$ by +\begin{equation}\label{def hlm} +M_{\mathcal{B},q}h(x):=\left(\sup_{B'\in \mathcal{B'}} \frac{1_{B'}(x)}{\mu(B')}\int _{B'} |h(y)|^q\, d\mu(y)\right)^\frac 1q\, . +\end{equation} +Define further $M_{\mathcal{B}}:=M_{\mathcal{B},1}$. + +\begin{prop}\label{prop hlm} + Let $\mathcal{B}$ be a finite collection of balls in $X$. +If for some $\lambda>0$ and some measurable function $h:X\to [0,\infty)$ we have +\begin{equation}\label{eq ball assumption} +\int_{B'} h(x)\, d\mu(x)\ge \lambda \mu(B') +\end{equation} + for each $B'\in \mathcal{B}$, + then + \begin{equation}\label{eq besico} +\lambda \mu(\bigcup \mathcal{B}) \le 2^{2a}\int_X h(x)\, d\mu(x)\, . + \end{equation} +\rs{For the union of balls, I would write $\bigcup_{B\in\mathcal{B}} B$, here and elsewhere} +For every measurable function $h$ \rs{Choose a different name for the function as $h$ is already used above} +and $1\le q'0}B(x_0,R), +\end{equation} because every point of $X$ +has finite distance from $x_0$. +\rs{Can we not take $x_0$ to be the point $o$? Since we anyways intersect the set $G$ with $B(o, 2R)$ later} +\begin{lemma}\label{lemmarcut} +For all integers $R>0$ + \begin{equation} \label{Rcut} + \int 1_{G\cap B(x_0,R)} +\sup_{1/R0$. Replacing +$G$ by $G\cap B(o,R)$ if necessary, it suffices to show +\eqref{Rcut} under the assumption that $G$ is supported in $B(o,R)$. We make this assumption. +For every $x\in G$, the domain of integration +in \eqref{TRR} is contained in $B(o,2R)$. +Replacing +$F$ by $F\cap B(o,2R)$ if necessary, it suffices to show +\eqref{Rcut} under the assumption that $F$ is supported in $B(o,2R)$. We make this assumption. + +With the definition \eqref{defks} of $K_s$ +and the partition of unity \eqref{eq psisum}, we write \eqref{TRR} as the sum of \rs{Operator name changed from $S_{\sigma_1,\sigma_2,\mfa}$ to $\Tilde{T}_{\sigma_1,\sigma_2,\mfa}$ to remove conflict with the scale $S$} +\begin{equation}\label{middles} +\Tilde{T}_{\sigma_1,\sigma_2,\mfa}f(x)=\sum_{\sigma_1 \le s\le \sigma_2} +\int K_s(x,y) f(y) e(\mfa(y)) \, \mathrm{d}\mu(y) +\end{equation} +and +\begin{equation}\label{boundarys} +\sum_{s=\sigma_1-2,\sigma_1-1, \sigma_2+1,\sigma_2+2} +\int_{R_1 < \rho(x,y) < R_2} K_s(x,y) f(y) e(Q(y)) \, + \mathrm{d}\mu(y), +\end{equation} +where $\sigma_1$ is the smallest integer such that $D^{\sigma_1-2}R_2>\frac 1{4D}$ and $\sigma_2$ +is the largest integer so that $D^{\sigma_2+2}R_1<\frac 12$. Here we restricted the summation index $s$ +by omitting the summands with $s<\sigma_1-2$ +or $s>s_2+2$ because for these summands the function $K_s$ vanishes on the domain of integration, and we have ommitted the restriction in the integral +in the summands in \eqref{middles} because in theses summands the support of $K_s$ is contained in +the set described by this restriction. + + + +By the triangle inequality, it suffices to estimate +versions of \eqref{Rcut} separately with $T_{R_1,R_2,\mfa}$ replaced by +\eqref{middles} and by each summand of \eqref{boundarys}. +The case \eqref{middles} follows immediately from the following lemma, where we use that if +$\frac 1R\le R_1\le R_2\le R$, then $\sigma_1,\sigma_2$ +as in \eqref{middles} are in an interval $[-S,S]$ for some +sufficiently large $S$ depending on $R$. + +\begin{lemma}\label{lemmascut} +For all integers $S>0$ + \begin{equation} \label{Scut} + \int 1_{G}(x) +\max_{-S<\sigma_1\le \sigma_20$. + +\begin{lemma}\label{lemmasqcut} +For all finite sets $\tilde{\Mf}\subset \Mf$ + \begin{equation} \label{Sqcut} + \int 1_{G}(x) +\max_{-S<\sigma_1\le \sigma_2-S$, +assume $x$ is in $I_1(y_m,-S)$ for $m=1,2$. +Then, for $m=1,2$, there is $z_m\in Y_{k-1}\cap B(y_m,D^k)$ with $x\in I_3(z_m,k-1)$. +Using \eqref{disji} inductively for $j=3$, we +conclude $z_1=z_2$. This implies that the balls +$B(y_1, D^k)$ and $B(y_2, D^k)$ intersect. By construction of $Y_k$, this implies $y_1=y_2$. +This proves \eqref{disji} for $j=1$, + +We next consider \eqref{disji} for $j=3$. +Assume $x$ is in $I_3(y_m,k)$ for $m=1,2$ and $y_m\in Y_k$. If $x$ is in $X_k$, then by definition +\eqref{definei3}, $x\in I_1(y_m,k)$ for $m=1,2$. +As we have already shown \eqref{disji} for $j=1$, +we conclude $y_1=y_2$. This completes the proof in +case $x\in X_k$ and we may assume $x$ is not in $X_k$. By definition \eqref{definei3}, $x$ is not +in $I_3(z,k)$ for any $z$ with $z-S$. Let $x$ be a point of $B(o, 4D^S-2D^k$). +By induction, there is $y'\in Y_{k-1}$ such that +$x\in I_3(y',k-1)$. Using the inductive statement +\eqref{squeezedyadic}, we obtain $x\in B(y',4D^{k-1})$. +As $D>4$, we have applying the triangle inequality with +the points, $o$, $x$, and $y'$ we obtain that $y'\in B(o, 4D^S-D^k)$. +By Lemma \ref{coverball}, $y'$ is in $B(y,2D^k)$ +for some $y\in Y_k$. It follows that $x\in I_2(y,k)$. +This proves \eqref{unioni} for $j=2$. + +We show \eqref{unioni} for $j=3$. +Let $x\in B(o, 4D^S-2D^k)$. In case $x\in X_k$, + then by definition of $X_k$ we have $x\in I_1(y,k)$ for some $y\in Y_k$ and thus $x\in I_3(y,k)$. We may thus assume $x\not\in X_k$. As we have already seen +\eqref{unioni} for $j=2$, + there is $y\in Y_k$ such that $x\in I_2(y,k)$. +We may assume this $y$ is minimal with respect ot the order in $Y_k$ +Then $x\in I_3(y,k)$. + This proves \eqref{unioni} for $j=3$. + +Next we show the first inclusion in \eqref{squeezedyadic}. +Let $x\in B(y,\frac 1{2}D^k)$. +As $I_1(y,k)\subset I_3(y,k)$, +it suffices to show $x\in I_1(y,k)$. +If $k=-S$, this follows immediately from +the assumption on $x$ and definition of $I_1$. +Assume $k>-S$. By inductive statement \eqref{unioni} +and $D>4$, there is a +$y'\in Y_{k-1}$ such that $x\in I_3(y',k-1)$. +By inductive statement \eqref{squeezedyadic}, +we conclude $x\in B(y',4D^{k-1})$. +By the triangle inequality with points $x$, $y$, $y'$ and $D>4$, we have +$y'\in B(y,D^k)$. It follows by definition +\eqref{defineij} that +$I_3(y',k-1)\subset I_1(y,k)$ and thus +$x\in I_3(1,k)$. This proves the first inclusion +in \eqref{squeezedyadic}. + + +We show the second inclusion in \eqref{squeezedyadic}. +Let $x\in I_3(y,k)$. As $I_1(y,k)\subset I_2(y,k)$ +directly from the definition \eqref{defineij}, +it follows by definition \eqref{definei3} that +$x\in I_2(y,k)$. By definition +\eqref{defineij}, there is $y'\in Y_{k-1}\cap B(y,2D^k)$ +with $x\in I_3(y',k-1)$. By induction, +$x\in B(y', 4D^1{k-1})$. By the triangle inequality +applied to the points $x,y',y$ and $D>4$, we conclude +$x\in B(y,4D^k)$. +This shows the second inclusion in \eqref{squeezedyadic} and completes the proof of the lemma. +\end{proof} + +\begin{lemma}\label{icoveri} +Let $-S\le l\le k\le S$ and +$y\in Y_k$. +We have +\begin{equation}\label{3coverdyadic} + I_3(y,k)\subset \bigcup_{y'\in Y_l} I_3(y',l)\, . +\end{equation} +\end{lemma} +\begin{proof} + +Let $-S\le l\le k\le S$ and $y\in Y_k$. +If $l=k$, the inclusion \eqref{3coverdyadic} +is true from the definition of set union. +We may then assume inductively that $k>l$ and the statement of the lemma is true if $k$ is replaced by $k-1$. +Let $x\in I_3(y,k)$. +By definition \eqref{definei3}, $x\in I_j(y,k)$ +for some $j\in \{1,2\}$. By \eqref{defineij}, +$x\in I_3(w,k-1)$ for some $w\in Y_{k-1}$. +We conclude \eqref{3coverdyadic} by induction. +\end{proof} + +\begin{lemma}\label{3dyadiclemma} +Let $-S\le l\le k\le S$ and +$y\in Y_k$ and $y'\in Y_l$. +with +$I_3(Y',l)\cap I_3(y,k)\neq \emptyset$. Then +\begin{equation} + \label{3dyadicproperty} +I_3(y',l)\subset I_3(y,k). +\end{equation} + + +\end{lemma} + +\begin{proof} +Let $l,k,y,y'$ as in the Lemma. +Pick $x\in I_3(y',l)\cap I_3(y,k)$. +Assume first $l=k$. By \eqref{disji} of Lemma +\ref{firstgridlemma}, we conclude $y'=y$ +and thus \eqref{3dyadicproperty}. +Now assume $l2A^6$ implies \eqref{new small boundary}. + +It thus remains to prove that the balls +occurring in +\eqref{sumcompare1} are pairwise disjoint. +Let $(u,l)$ and $(u',l')$ be two parameter pairs occurring +in the sum of \eqref{sumcompare1} and let +$ B(u, \frac 14 D^l)$ and $B({u'}, \frac 14 D^{l'})$ +be the corresponding balls. If +$l=l'$, then the balls are equal or disjoint by +\eqref{squeezedyadic} and \eqref{disji} of Lemma \ref{firstgridlemma}. Assume then without loss of generality that $l'32$, +\begin{equation}\label{bulbul} + B(u', \frac 13 D^{l'})\subset B(u, \frac 38 D^l). +\end{equation} +As $(u',l'|y,k)$, there is a point $x$ in +$X\setminus I_3(y,k)$ with $\rho(x,u')<4D^{k'}$. +Using $D>32$, we conclude from \eqref{bulbul} that +$x\in B(u,\frac 12D^l)$. However, $B(u,\frac 12 D^l)\subset I_3(u,l)$, +and $I_3(u,l)\subset I_3(y,k)$, a contradiction to +$x\not\in I_3(y,k)$. +This proves the lemma. +\end{proof} + + +\begin{lemma} +Let $K$ be the smallest integer larger than $2A^6$ +and let $n\ge 0$ be an integer. Then +for each $-S+nK\le k\le S$ we have + \begin{equation} + \label{very new small} + \sum_{y'\in Y_{k-nK}: (y',k-nK|y,k)}\mu(I_3(y',k-nK)) \le 2^{-n} \mu(I_3(y,k))\,. + \end{equation} +\end{lemma} +\begin{proof} + We prove this by induction on $n$. If $n=0$, + both sides of \eqref{very new small} are equal to + $\mu(I_3(y,k)$. If $n=1$, this follows from lemma \ref{new small boundary}. + + Assume $n>1$ and \eqref{very new small} has been proven with $n-1$ in place of $n$. +We write \eqref{very new small} + \begin{equation} + \sum_{y''\in Y_{k-nK}: (y'',k-nK|y,k)}\mu(I_3(y'',k-nK)) + \end{equation} + \begin{equation} += \sum_{y'\in Y_{k-K}:(y',k-K|y,k)} \left[ \sum_{y''\in Y_{k-nK}: (y'',k-nK|y',k-K)}\mu(I_3(y'',k-nK)) \right] + \end{equation} +Applying the induction hypothesis, this is bounded by + \begin{equation} += \sum_{y'\in Y_{k-K}:(y',k-K|y,k)} 2^{1-n}\mu(I_3(y',k-K)) + \end{equation} +Applying Lemma \ref{new small boundary} gives +\eqref{very new small} and proves the lemma. +\end{proof} + +\begin{lemma} + For each $-S\le k\le S$ and $y\in Y_k$ and $0D^{-S}$ we have + \begin{equation} + \label{old small boundary} + \mu(\{x \in I(y,k) \ : \ \rho(x, X \setminus I(y,k)) \leq t D^{k}\}) \le 4A^2 t^\kappa \mu(I)\,. + \end{equation} +\end{lemma} +\begin{proof} +Let $x\in I(y,k)$ with $\rho(x, X \setminus I(y,k)) \leq t D^{k}$. Let + +\end{proof} + + + + + + + + + + + + + +Let $\mathcal{D}$ the set of all $I_3(x,k)$ with $k\in [-S,S]$ and +$y\in Y_k$. Define +\begin{equation} +s(I_3(x,k)):=k +\end{equation} +\begin{equation} +c(I_3(y,k)):=y +\end{equation} +We show that $(\mathcal{D},c,s)$ constitutes a grid structure. Property \eqref{eq vol sp cube} +follows from \eqref{squeezedyadic}. +To see \eqref{coverdyadic}, let $I\in \mathcal{D}$ +and $-S\le k< s(I)$. Let $x\in I$. + + +We show properties +\eqref{coverdyadic}, +\eqref{dyadicproperty}, +\eqref{eq vol sp cube}, and \eqref{eq small boundary} +for this $\mathcal {D}$, $s$, and $x$. + +We first show \eqref{coverdyadic}. +Assume to get a contradiction that \eqref{coverdyadic} +is false. Then there is a $I$ violating the conclusion of +\eqref{coverdyadic}. Pick such $I=I(x,l)$ such that $l$ is minimal. +Let $-S\le kk$. Choose $y\in I\cap J$. By property \eqref{coverdyadic}, +there is $K\in \mathcal{D}$ with $s(K)=s(J)-1$ with $y\in K$. By construction +of $J$, and pairwise disjointness of all $I(w,s(J)-1)$ that we have already seen, +we have $K\subset J$. By minimality if $s(J)$, we have $I\subset K$. +This proves $I\subset J$ and thus \eqref{dyadicproperty}. + + +\section{Proof of L.\ref{tilelemma}, tile structure} +\label{subsectiles} + +\begin{lemma} + \label{lem d monotone} + Let $\mathcal{E} \subset \mathcal{E}' \subset X$. Then it holds for all $\theta, \vartheta \in \Theta$ that + $$ + d_{\mathcal{E}}(\theta, \vartheta) \le d_{\mathcal{E}'}(\theta, \vartheta)\,. + $$ +\end{lemma} + +\begin{proof} + This follows immediatly from the definition \eqref{definedE} and monotonicity of suprema with respect to set inclusion. +\end{proof} + +Choose a grid structure $(\mathcal{D}, c, s)$. For cubes $I \in \mathcal{D}$, we will denote +$$ + I^\circ = B(c(I), \frac{1}{4} D^{s(I)})\,. +$$ + +\lars{The definitions of the doubling roperties changed, the cubes do not have to have the same centers. Check if this is easier to prove now.} +\begin{lemma} + \label{lem cube monotone} + Let $I, J \in \mathcal{D}$ with $I \subset J$. + Then for all $\theta, \vartheta \in\Theta$ we have + $$ + d_{I^\circ}(\theta, \vartheta) \le d_{J^\circ}(\theta, \vartheta)\,, + $$ + and if $I \ne J$ then we have + $$ + d_{I^\circ}(\theta, \vartheta) \le 2^{-95a} d_{J^\circ}(\theta, \vartheta)\,. + $$ +\end{lemma} + +\begin{proof} + If $s(I) \ge s(J)$ then \eqref{dyadicproperty} and the assumption $I\subset J$ imply $I = J$. Then the Lemma holds by reflexivity. + + If $s(J) \ge s(I)+1$, then using Lemma \ref{lem d monotone}, \eqref{defineD} and \eqref{seconddb}, we get + \begin{equation} + \label{eq dIJ est} + d_{I^\circ}(\theta, \vartheta) \le d_{B(c(I), 4 D^{s(I)})}(\theta, \vartheta) \le 2^{-100a} d_{B(c(I), 4D^{s(J)})}(\theta, \vartheta)\,. + \end{equation} + Using \eqref{eq vol sp cube}, together with the inclusion $I \subset J$, we obtain + $$ + c(I) \in I \subset J \subset B(c(J), 4 D^{s(J)}) + $$ + and consequently by the triangle inequality + $$ + B(c(I), 4 D^{s(J)}) \subset B(c(J), 8 D^{s(J)})\,. + $$ + Using this together with Lemma \ref{lem d monotone} and \eqref{firstdb} in \eqref{eq dIJ est}, we obtain + \begin{align*} + d_{I^\circ}(\theta, \vartheta) &\le 2^{-100a} d_{B(c(J), 8D^{s(J)})}(\theta, \vartheta)\\ + &\le 2^{-100a + 5a} d_{B(c(J), \frac{1}{4}D^{s(J)})}(\theta, \vartheta)\\ + &= 2^{-95a}d_{J^\circ}(\theta, \vartheta)\,. + \end{align*} + This proves the second inequality claimed in the Lemma, from which the first follows since $a \ge 4$ and hence $2^{-95a} \le 1$. +\end{proof} + +\begin{lemma} + \label{lem tile center 1} + Let $I \in \mathcal{D}$. Let $Z \subset \Theta$ be such that + \begin{equation} + \label{eq tile Z} + Z \subset \bigcup_{q \in Q(X)} B_{I^\circ}(q, 1) + \end{equation} + and for any $z, z' \in Z$ we have + \begin{equation} + \label{eq tile disjoint Z} + B_{I^\circ}(z, 0.3) \cap B_{I^\circ}(z', 0.3) = \emptyset\,. + \end{equation} + Then the cardinality of $Z$ is estimated by + $$ + 2^{2a}|Q(X)|\,. + $$ +\end{lemma} + +\begin{proof} + By applying property \eqref{thirddb} $2$ times, we obtain for each ball $B_{I^\circ}(q,1)$ a collection $J(q) \subset \Theta$ such that $|J(q)| \le 2^{2a}$ and + $$ + B_{I^\circ}(q,1) \subset \bigcup_{j \in J(q)} B_{I^\circ{}}(j, \frac{1}{4})\,. + $$ + Thus for each $z \in Z$, there exists a $j(z) \in \bigcup_{q \in Q(x)} J(q)$ such that $d_{I^\circ}(z,j(z)) < \frac{1}{4} <0.3$ and hence $j(z) \in B_{I^\circ}(z, 0.3)$. By the assumption \eqref{eq tile disjoint Z}, it follows that the map $z \mapsto j(z)$ is injective. This establishes the Lemma, since then + $$ + |Z| \le |\bigcup_{q \in Q(x)} J(q)| \le \sum_{q \in Q(X)}|J(q)| \le \sum_{q \in Q(X)} 2^{2a} \le |Q(X)|2^{2a}\,. + $$ +\end{proof} + +By Lemma \ref{lem tile center 1}, for each $I \in \mathcal{D}$, there exists a set $Z(I)$ satisfying \eqref{eq tile Z} and \eqref{eq tile disjoint Z}, such that $Z(I)$ has maximal cardinality among all such sets. We pick for each $I \in \mathcal{D}$ such a set $Z(I)$. + +\begin{lemma} + For each $I \in \mathcal{D}$, we have + \begin{equation} + \label{eq tile cover} + Q(X) \subset \bigcup_{q \in Q(X)} B_{I^\circ}(q, 1) \subset \bigcup_{z \in Z(I)} B_{I^\circ}(z, 0.7)\,. + \end{equation} +\end{lemma} + +\begin{proof} + To show \eqref{eq tile cover} note that the first inclusion is obvious. For the second inclusion let $q' \in \bigcup_{q \in Q(X)} B_{I^\circ}(q, 1)$. By maximality of $Z(I)$, there must be a point $z \in Z(I)$ such that $B_{I^\circ}(z, 0.3) \cap B_{I^\circ}(q', 0.3) \ne \emptyset$. Else, $Z(I) \cup \{q'\}$ would be a set of larger cardinality than $Z(I)$ satisfying \eqref{eq tile Z} and \eqref{eq tile disjoint Z}. Fix such $z$, and fix a point $z_1 \in B_{I^\circ}(z, 0.3) \cap B_{I^\circ}(q', 0.3)$. By the triangle inequality, we deduce that + $$ + d_{I^\circ}(z,q') \le d_{I^\circ}(z,z_1) + d_{I^\circ}(q', z_1) < 0.3 + 0.3 = 0.6\,, + $$ + and hence $q' \in B_{I^\circ}(z, 0.7)$. +\end{proof} + +We define +$$ + \fP = \{(I, z) \ : \ I \in \mathcal{D}, z \in Z(I)\}\,. +$$ +We define $\sc((I, z)) = I$ and $\fcc((I, z)) = z$. We further set $s(\fp) = s(\sc(\fp))$, $c(\fp) = c(\sc(\fp))$. Then \eqref{tilecenter}, \eqref{tilescale} hold. + +It remains to construct the map $\Omega$. We first construct an auxilliary map $\Omega_1$. For each $I \in \mathcal{D}$, we pick an enumeration of the finite set $Z(I)$ +$$ + Z(I) = \{z_1, \dotsc, z_M\}\,. +$$ +We define +$$ + \Omega_1((I, z_1)) = B_{I^\circ}(z_1, 0.7) \setminus \bigcup_{z \in Z(I)} B_{I^\circ}(z, 0.3)\,, +$$ +and then we iteratively define +\begin{equation} + \label{eq def omega1} + \Omega_1((I, z_k)) = B_{I^\circ}(z_k, 0.7) \setminus \bigcup_{z \in Z(I) \setminus \{z_k\}} B_{I^\circ}(z, 0.3) \setminus \bigcup_{i=1}^{k-1} \Omega_1((I, z_i))\,. +\end{equation} + +\begin{lemma} + \label{lem omega1 disj} + For each $I \in \mathcal{D}$, the sets $\Omega_1(\fp), \fp \in \fP(I)$ are pairwise disjoint. +\end{lemma} + +\begin{proof} + By the definition of the map $\sc$, we have + $$ + \fP(I) = \{(I, z) \, : \, z \in Z(I)\}\,. + $$ + By \eqref{eq def omega1}, the set $\Omega_1((I, z_k))$ is disjoint from each $\Omega_1((I, z_i))$ with $i < k$. Thus the sets $\Omega_1(\fp)$, $\fp \in \fP(I)$ are pairwise disjoint. +\end{proof} + +\begin{lemma} + For each $I \in \mathcal{D}$, it holds that + \begin{equation} + \label{eq omega1 cover} + \bigcup_{z \in Z(I)} B_{I^\circ}(z, 0.7)\subset \bigcup_{\fp \in \fP(I)} \Omega_1(\fp)\,. + \end{equation} + For every $\fp \in \fP$, it holds that + \begin{equation} + \label{eq omega1 incl} + B_{\fp}(\fcc(\fp), 0.3) \subset \Omega_1(\fp) \subset B_{\fp}(\fcc(\fp), 0.7)\,. + \end{equation} +\end{lemma} + +\begin{proof} + For \eqref{eq omega1 incl} let $\fp = (I, z)$. + The second inclusion in \eqref{eq omega1 incl} then follows from \eqref{eq def omega1} and the equality $B_{\fp}(\fcc(\fp), 0.7) = B_{I^\circ}(z, 0.7)$, which is true by definition. + For the first inclusion in \eqref{eq omega1 incl} let $q \in B_{\fp}(\fcc(\fp),0.3)$. Let $k$ be such that $z = z_k$ in the enumeration we chose above. It follows immediately from \eqref{eq def omega1} and \eqref{eq tile disjoint Z} that + $q \notin \Omega_1((I, z_i))$ for all $i < k$. Thus, again from \eqref{eq def omega1}, we have + $q \in \Omega_1((I,z_k))$. + + To show \eqref{eq omega1 cover} let $q \in \bigcup_{z \in Z(I)} B_{I^\circ}(z,0.7)$. + If there exists $z \in Z_1(I)$ with $q \in B_{I^\circ}(z,0.3)$, then + $$ + z \in \Omega_1((I, z)) \subset \bigcup_{\fp \in \fP(I)} \Omega_1(\fp) + $$ + by the first inclusion in \eqref{eq omega1 incl}. + + Now suppose that there exists no $z \in Z(I)$ with $q \in B_{I^\circ}(z, 0.3)$. Let $k$ be minimal such that $q \in B_{I^\circ}(z_k, 0.7)$. Since $\Omega_1((I, z_i)) \subset B_{I^\circ}(z_i, 0.7)$ for each $i$ by \eqref{eq def omega1}, we have that $q \notin \Omega_1((I, z_i))$ for all $i < k$. Hence $q \in \Omega_1((I, z_k))$, again by \eqref{eq def omega1}. +\end{proof} + +Now we are ready to define the function $\Omega$. For all cubes $I \in \mathcal{D}$ such that there exists no $J \in \mathcal{D}$ with $I \subset J$ and $I \ne J$, we define for all $\fp \in \fP(I)$ +\begin{equation} + \label{eq max omega} + \fc(\fp) = \Omega_1(\fp)\,. +\end{equation} +For cubes $I \in \mathcal{D}$ for which there exists $J \in \mathcal{D}$ with $I \subset J$ and $I \ne J$, we define $\Omega$ by recursion. We can pick an inclusion minimal $J \in \mathcal{D}$ among the finitely many cubes such that $I \subset J$ and $I \ne J$. This $J$ is unique: Suppose that $J'$ is another inclusion minimal cube with $I \subset J'$ and $I \ne J'$. Without loss of generality, we have that $s(J) \le s(J')$. By \eqref{dyadicproperty}, it follows that $J \subset J'$. Since $J'$ is minimal with respect to inclusion, it follows that $J = J'$. Then we define +\begin{equation} + \label{eq it omega} + \fc(\fp) = \bigcup_{z \in Z(J) \cap \Omega_1(\fp)} \Omega((J, z)) \cup B_{\fp}(\fcc(\fp),0.2)\,. +\end{equation} +%\lars{Also, the definition of dyadic cubes does not exclude that there are some small cubes and all their children of scale $\ge -S$, disjoint from $B(o,D^S)$, right?} +%\ct{I have just done something like parent in Lemma \ref{lem antichain 0}. This is however in flux. As a rule of thump I am not trying to have too many definitions, so I did not +%define parent. Such definitions just lead to crazy referencing around. Sometimes a few lines of code, even if repeated 2-3 times throughout text are better. Ofcourse +%it all depends... And yes, the limit at +%plusminus S causes some case distinctions} +%\lars{this is not just about the limit at plusminus $S$, but also that a maximal cube does not necessarily have scale $S$. That causes no problems here, I just wanted to point it out} +%\ct{OK, I see. Probably not an issue, as we later always have reference points in B(o,S) when we +%use something like coverball.This is more o} + + +\begin{lemma} + With this definition, \eqref{eq dis freq cover}, \eqref{eq freq dyadic} and \eqref{eq freq comp ball} hold. +\end{lemma} + +\begin{proof} + First, we prove \eqref{eq freq comp ball}. If $I \in \mathcal{D}$ is maximal in $\mathcal{D}$ with respect to set inclusion, then \eqref{eq freq comp ball} holds for all $\fp \in \fP(I)$ by \eqref{eq max omega} and\eqref{eq omega1 incl}. Now suppose that $I$ is not maximal in $\mathcal{D}$ with respect to set inclusion. Then we may assume by induction that for all $J \in \mathcal{D}$ with $I \subset J$ and all $\fp' \in \fP(J)$, \eqref{eq freq comp ball} holds. Let $J$ be the unique minimal cube in $\mathcal{D}$ with $I \subsetneq J$. + + Suppose that $q \in \Omega(\fp)$. By \eqref{eq it omega}, there exists $z \in Z(J) \cap \Omega_1(\fp)$ with $q \in \Omega(J,z)$. Using the triangle inequality and \eqref{eq omega1 incl}, we obtain + $$ + d_{I^\circ}(\fcc(\fp),q) \le d_{I^\circ}(\fcc(\fp), z) + d_{I^\circ}(z, q) \le 0.7 + d_{I^\circ}(z, q)\,. + $$ + By Lemma \ref{lem cube monotone} and the induction hypothesis, this is estimated by + $$ + \le 0.7 + 2^{-95a} d_{J^\circ}(z,q) \le 0.7 + 2^{-95a}\cdot 1 < 1\,. + $$ + This shows the second inclusion in \eqref{eq freq comp ball}. The first inclusion is immediate from \eqref{eq it omega}. + + Next, we show \eqref{eq dis freq cover}. Let $I \in \mathcal{D}$. + + If $I$ is maximal with respect to inclusion, then disjointness of the sets $\fc(\fp)$ for $\fp \in \fP(I)$ follows from the definition \eqref{eq max omega} and Lemma \ref{lem omega1 disj}. To obtain the inclusion in \eqref{eq dis freq cover} one combines the inclusions \eqref{eq tile cover} and \eqref{eq omega1 cover} with \eqref{eq max omega}. + + Now we turn to the case where there exists $J \in \mathcal{D}$ with $I \subset J$ and $I\ne J$. In this case we use induction: It suffices to show \eqref{eq dis freq cover} under the assumption that it holds for all cubes $J \in \mathcal{D}$ with $I \subset J$. As shown before definition \eqref{eq it omega}, we may choose the unique inclusion minimal such $J$. To show disjointness of the sets $\fc(\fp), \fp \in \fP(I)$ we pick two tiles $\fp, \fp' \in \fP(I)$ and $x \in \fc(\fp) \cap \fc(\fp')$. + Then we are by \eqref{eq it omega} in one of the following four cases. + + 1. There exist $z \in Z(J) \cap \Omega_1(\fp)$ such that $x \in \Omega(J, z)$, and there exists $z' \in Z(J) \cap \Omega_1(\fp')$ such that $x \in \Omega(J, z')$. By the induction hypothesis, that \eqref{eq dis freq cover} holds for $J$, we must have $z = z'$. By Lemma \ref{lem omega1 disj}, we must then have $\fp = \fp'$. + + 2. There exists $z \in Z(J) \cap \Omega_1(\fp)$ such that $x \in \Omega(J,z)$, and $x \in B_{\fp'}(\fcc(\fp'), 0.2)$. Using the triangle inequality, Lemma \ref{lem cube monotone} and \eqref{eq freq comp ball}, we obtain + $$ + d_{\fp'}(\fcc(\fp'),z) \le d_{\fp'}(\fcc(\fp'), x) + d_{\fp'}(z, x) \le 0.2 + 2^{-95a} \cdot 1 < 0.3\,. + $$ + Thus $z \in \Omega_1(\fp')$ by \eqref{eq omega1 incl}. By Lemma \ref{lem omega1 disj}, it follows that $\fp = \fp'$. + + 3. There exists $z' \in Z(J) \cap \Omega_1(\fp')$ such that $x \in \Omega(J,z')$, and $x \in B_{\fp}(\fcc(\fp), 0.2)$. This case is the same as case 2., after swapping $\fp$ and $\fp'$. + + 4. We have $x \in B_{\fp}(\fcc(\fp), 0.2) \cap B_{\fp'}(\fcc(\fp'), 0.2)$. In this case it follows that $\fp = \fp'$ since the sets $B_{\fp}(\fcc(\fp), 0.2)$ are pairwise disjoint by the inclusion \eqref{eq omega1 incl} and Lemma \ref{lem omega1 disj}. + + To show the inclusion in \eqref{eq dis freq cover}, let $q \in Q(X)$. By the induction hypothesis, there exists $\fp \in \fP(J)$ such that $q \in \Omega(\fp)$. By definition of the set $\fP$, we have $\fp = (J, z)$ for some $z \in Z(J)$. By \eqref{eq tile Z}, there exists $x \in X$ with $d_{J^\circ}(Q(x), z) \le 1$. By Lemma \ref{lem cube monotone}, it follows that $d_{I^\circ}(Q(x), z) \le 1$. + Thus, by \eqref{eq tile cover}, there exists $z' \in Z(I)$ with $z \in B_{I^\circ}(z', 0.7)$. Then by \eqref{eq omega1 cover} there exists $\fp' \in \fP(I)$ with $z \in Z(J) \cap \Omega_1(\fp')$. Consequently, by \eqref{eq it omega}, $q \in \fc(\fp')$. This completes the proof of \eqref{eq dis freq cover}. + + Finally, we show \eqref{eq freq dyadic}. Let $\fp, \fq \in \fP$ with $\sc(\fp) \subset \sc(\fp)$ and $\fc(\fp) \cap \fc(\fq) \ne \emptyset$. If we have $s(\sc(\fp)) \ge s(\sc(\fq))$, then it follows from \eqref{dyadicproperty} that $I = J$, thus $\fp, \fq \in \fP(I)$. By \eqref{eq dis freq cover} we have then either $\fc(\fp) \cap \fc(\fq) = \emptyset$ or $\fc(\fp) = \fc(\fq)$. By the assumption in \eqref{eq freq dyadic} we have $\fc(\fp) \cap \fc(\fq) \ne \emptyset$, so we must have $\fc(\fp) = \fc(\fq)$ and in particular $\fc(\fq) \subset \fc(\fp)$. + + So it remains to show \eqref{eq freq dyadic} under the additional assumption that $s(\sc(\fq)) > s(\sc(\fp))$. In this case, we argue by induction on $s(\sc(\fq))-s(\sc(\fp))$. By \eqref{coverdyadic}, there exists a cube $J \in \mathcal{D}$ with $s(J) = s(\sc(\fq)) - 1$ and $J \cap\sc(\fp) \ne \emptyset$. We pick one such $J$. By \eqref{dyadicproperty}, we have $\sc(\fp) \subset J \subset \sc(\fq)$. + + By \eqref{eq tile Z}, there exists $x \in X$ with $d_{\fq}(Q(x), \fcc(\fq)) \le 1$. By Lemma \ref{lem cube monotone}, it follows that $d_{J^\circ}(Q(x), \fcc(\fq)) \le 1$. + Thus, by \eqref{eq tile cover}, there exists $z' \in Z(J)$ with $\fcc(\fq) \in B_{J^\circ}(z', 0.7)$. Then by \eqref{eq omega1 cover} there exists $\fq' \in \fP(J)$ with $\fcc(\fq) \in\Omega_1(\fq')$. + By \eqref{eq it omega}, it follows that $\Omega(\fq) \subset \Omega(\fq')$. Note that then $\sc(\fp) \subset \sc(\fq')$ and $\fc(\fp) \cap \fc(\fq') \ne \emptyset$ and $s(\fq') - s(\fp) = s(\fq) - s(\fp) - 1$. Thus, we have by the induction hypothesis that $\Omega(\fq') \subset \Omega(\fp)$. This completes the proof. +\end{proof} + +\chapter{P. \ref{prop dyadic} from forest P. \ref{antichainprop} and antichain P. \ref{forestprop}} +\label{proptopropprop} + +%\section{Decomposition of grid cubes and first exceptional set} + + +Let a grid structure $(\mathcal{D}, c, s)$ and a tile structure $(\fP,\sc,\fc,\fcc)$ +for this grid structure be given. +In Subsection \ref{subsectilesorg}, we decompose the +set $\fP$ of tiles into subsets. +Each subset will be controlled by one of three methods. +The guiding principle of the decomposition is +to be able to apply the forest estimate +of Proposition \ref{forestprop} to the final subsets +defined in \eqref{defc5}. This application is done in Subsection \ref{subsecforest}. +The miscellaneous subsets along the construction of the +forests will be either thrown into exceptional sets, +which are defined and controlled in Subsection +\ref{subsetexcset}, or will be controlled by +the antichain estimate of Proposition \ref{antichainprop}, +which is done in Subsection \ref{subsecantichain}. +%The oganisation and summation of these various +%estimates is done in Subsection \ref{subsecaddingup}. + + + + +\section{Organisation of the tiles}\label{subsectilesorg} + +In the following chain of definitions, $k, n$, and +$j$ will be nonnegative integers. + +Define +$\mathcal{C}(G,k)$ to be the set of $I\in \mathcal{D}$ +such that there exists a $J\in \mathcal{D}$ with $I\subset J$ +and +\begin{equation}\label{muhj1} + {\mu(G \cap J)} > 2^{-k-1}{\mu(G)}\, , +\end{equation} +but there does not exists a $J\in \mathcal{D}$ with $I\subset J$ and +\begin{equation}\label{muhj2} + {\mu(G \cap J)} > 2^{-k}{\mu(J)}\,. +\end{equation} +Let +\begin{equation} + \label{eq defPk} + \fP(k)=\{\fp\in \fP \ : \ \sc(\fp)\in \mathcal{C}(G,k)\} +\end{equation} +Define $ {\mathfrak{M}}(n,k)$ to be the set of $\fp \in \fP(k)$ such that + \begin{equation}\label{ebardense} + \mu({E_1}(\fp)) > 2^{-n} \mu(\sc(\fp)) + \end{equation} +and there does not exists $\fp'\in \fP(k)$ with +$\fp'\neq \fp$ and $\fp\le \fp'$ such that + \begin{equation}\label{mnkmax} + \mu({E_1}(\fp')) > 2^{-n} \mu(\sc(\fp')). + \end{equation} +Define for a collection $\fP'\subset \fP(k)$ +\begin{equation} + \label{eq densdef} + \dens_k' (\fP'):= \sup_{\fp'\in \fP'}\sup_{\lambda \geq 2} \lambda^{-a} \sup_{\fp \in \fP(k): \lambda \fp' \lesssim \lambda \fp} + \frac{\mu({E}_2(\lambda, \fp))}{\mu(\sc(\fp))}\,. +\end{equation} +Sorting by density, we define +\begin{equation} + \label{def cnk} + \fC(n,k):=\{\fp\in \fP(k) \ : \ + 2^{4a}2^{-n}< \dens_k'(\{\fp\}) \le + 2^{4a}2^{-n+1}\}\,. +\end{equation} +Following Fefferman \cite{fefferman}, we +define for $\fp \in \fP(k)$ + \begin{equation}\label{defbfp} + \mathfrak{B}(\fp) := \{ \mathfrak{m} \in \mathfrak{M}(n,k) \ : \ 2 \fp \lesssim 100 \mathfrak{m}\} + \end{equation} +and +\begin{equation}\label{defcnkj} + \fC_1(n,k,j) := \{\fp \in \fC(n,k) \ : \ 2^{j} \leq |\mathfrak{B}(\fp)| < 2^{j+1}\}\,. +\end{equation} +and +\begin{equation}\label{defl0nk} + \fL_0(n,k) := \{\fp \in \fC(n,k) \ : \ |\mathfrak{B}(\fp)| <1\}\,. +\end{equation} +Together with the following removal of minimal layers, the splitting into $\fC_1(n,k,j)$ will lead to a separation of trees. +Define recursively for $0\le l\le Z(n+1)$ +\begin{equation} + \label{eq L1 def} + \fL_1(n,k,j,l) +\end{equation} +to be the set of minimal elements with respect to $\le$ in +\begin{equation} + \fC_1(n,k,j)\setminus \bigcup_{0\le l' \mu(G)\,. +\end{equation} +Define $\fP_{F,G}$ to be the set of all $\fp\in \fP$ +with $\dens_2(\{\fp\})\ge 2^{-k_0}$. Define +\begin{equation}\label{definegone} + G_1:=\bigcup_{\fp\in \fP_{F,G} }\sc(\fp)\, . +\end{equation} +For an integer $\lambda\ge 0$, define $A(\lambda,n,k)$ to be the set of all $x\in X$ such that +\begin{equation} + \label{eq Aoverlap def} + \sum_{\fp \in \mathfrak{M}(n,k)}1_{\sc(\fp)}(x)>1+\lambda 2^{n+1} +\end{equation} +and define +\begin{equation}\label{definegone2} + G_2:= +\bigcup_{0\le k}\bigcup_{k< n} +A(2n+6,k,n)\, . +\end{equation} +Define + \begin{equation}\label{defineg3} + G_3 := + \bigcup_{k\ge 0}\, \bigcup_{n \geq k}\, + \bigcup_{0\le j\le 2n+3} + \bigcup_{\fp \in \fL_4 (n,k,j)} + \sc(\fp)\, . + \end{equation} +Define $G'=G_1\cup G_2 \cup G_3$ +The following bound of the measure of $G'$ will be proven in +Subsection \ref{subsetexcset}. +\begin{lemma}\label{allgbound} +We have +\begin{equation} + \mu(G')\le 2^{-2}\mu(G)\, . +\end{equation} +\end{lemma} + +In Subsection \ref{subsecforest}, we identify each set $\fC_5(k,n,j)$ as forest and use Proposition +\ref{forestprop} to prove the following lemma. + +\lars{Add constants in the following two lemmas, after figuring out the exact constants in tree section} + +\begin{lemma}\label{subsecflemma} + Let + \begin{equation} + \fP'=\bigcup_{k\ge 0}\bigcup_{n\ge k} + \bigcup_{0\le j\le 2n+3}\fC_5(k,n,j) + \end{equation} + For all $f:X\to \C$ with $|f|\le \mathbf{1}_F$ we have +\begin{equation} + \label{disclesssim1} + \int_{G \setminus G'} \left|\sum_{\fp \in \fP'} T_{\fp} f \right|\, \mathrm{d}\mu \lesssim \mu(G)^{1/q'} \mu(F)^{1/q}\,. +\end{equation} +\end{lemma} + +In Subsection \ref{subsecforest}, we decompose +the complement of the set of tiles in Lemma +\ref{subsecflemma} and apply the antichain estimate of +Proposition \ref{antichainprop} to prove the following lemma. + +\begin{lemma}\label{subsecalemma} + Let + \begin{equation} + \fP'=\fP\setminus \left(\bigcup_{k\ge 0}\bigcup_{n\ge k} + \bigcup_{0\le j\le 2n+3}\fC_5(k,n,j)\right)\,. + \end{equation} + For all $f:X\to \C$ with $|f|\le \mathbf{1}_F$ we have +\begin{equation} + \label{disclesssim2} + \int_{G \setminus G'} \left|\sum_{\fp \in \fP'} T_{\fp} f\right| \, \mathrm{d}\mu \lesssim \mu(G)^{1/q'} \mu(F)^{1/q}\,. +\end{equation} +\end{lemma} +Proposition \ref{prop dyadic} follows by applying +triangle inequality to \eqref{disclesssim} +according to the splitting in Lemma \ref{subsecflemma} +and \ref{subsecalemma} and using both Lemmas as well +as the bound on the set $G'$ given by Lemma \ref{allgbound}. + + + +\section{Proof of Lemma \ref{allgbound}, the exceptional sets} +\label{subsetexcset} + + +We prove separate bounds for $G_1$, $G_2$, and $G_3$ +in Lemmas \ref{g1bound}, +\ref{g2bound}, and \ref{g3bound}. Adding up these bounds proves Lemma \ref{allgbound}. + +The bound for $G_1$ is follows from the Besicovitch covering lemma, Proposition \ref{prop hlm}. + +\begin{lemma}\label{g1bound} +We have +\begin{equation} + \mu(G_1)\le 2^{-4}\mu(G)\, . +\end{equation} +\end{lemma} +\begin{proof} +For each $\fp\in \fP_{F,G}$ pick a +$r(\fp)>4D^{\ps(\fp)}$ with +$$ + {\mu(F\cap B(\pc(\fp),r(\fp)))}\ge 2^{-k_0-1}{\mu(B(\pc(\fp),r(\fp)))}\, . +$$ +This ball exists by definition of $\fP_{F,G}$ +and $\dens_2$. By applying Proposition \ref{prop hlm} to the collection of balls +$$ + \mathcal{B} = \{B(\pc(\fp),r(\fp)) \ : \ \fp \in \fP_{F,G}\} +$$ +and the function $h = 1_F$, we obtain +$$ + \mu(\bigcup \mathcal{B}) \le 2^{2a} 2^{k_0 +1} \mu(F)\,. +$$ +We conclude with \eqref{eq vol sp cube} and $r(\fp)>4D^{\ps(\fp)}$ +$$ + \mu(G_1)\le \mu(\bigcup_{\fp\in \fP_{F,G}} \sc(\fp)) + \le \mu(\bigcup \mathcal{B})\le 2^{2a} 2^{k_0 + 1} \mu (F)\,. +$$ +Using the definition of $k_0$, this proves the lemma. +\end{proof} + + +We turn to the bound of $G_2$, which relies on the dyadic covering Lemma \ref{ckmeasure} and the +John-Nirenberg Lemma \ref{jn} below. + +\begin{lemma}\label{ckmeasure} +For each $k\ge 0$, the union of all intervals +in $\mathcal{C}(G,k)$ has measure at most $2^{k+1} \mu(G)$ . +\end{lemma} +\begin{proof} + The union of intervals in $\mathcal{C}(G,k)$ +is contained the union of the set $\mathcal{M}(k)$ +of all intervals $J$ with +${\mu(G \cap I)} > 2^{-k-1}{\mu(I)}$. +The set $\mathcal{M}(k)$ is contained in the union of +the set $\mathcal{M}^*(k)$ of maximal elements in +$\mathcal{M}(k)$ with respect to set inclusion. Hence +\begin{equation}\label{cbymstar} +\mu (\bigcup \mathcal{C}(G,k))\le \mu (\bigcup \mathcal{M}^*(k))\le +\sum_{J\in \mathcal{M}^*(k)}\mu(J) +\end{equation} +Using the definition of $\mathcal{M}(k)$ and then +the pairwise disjointness of elements in +$\mathcal{M}^*(k)$ \ct{this could be +an alternative defining property of dyadic intervals}, +we estimate \eqref{cbymstar} by +\begin{equation} +\le +2^{k+1}\sum_{J\in \mathcal{M}^*(k)}\mu(J\cap G) +\le 2^{k+1}\mu(G). +\end{equation} +This proves the lemma. +\end{proof} + + + + +\begin{lemma}\label{pairwise disjoint} + If $\fp, \fp' \in {\mathfrak{M}}(n,k)$ and + \begin{equation}\label{eintersect} + {E_1}(\fp)\cap {E_1}(\fp')\neq \emptyset, + \end{equation} + then $\fp=\fp'$. +\end{lemma} +\begin{proof} +Let $\fp,\fp'$ as in the lemma. As by definition of $E_1$ +we have +$E_1(\fp)\subset \sc(\fp)$ and analoguously for $\fp'$, we conclude from \eqref{eintersect} that $\sc(\fp)\cap \sc(\fp')\neq \emptyset$. Let without loss of generality $\sc(\fp)$ be maximal in +$\{\sc(\fp),\sc(\fp')\}$, then $\sc(\fp')\subset \sc(\fp)$. +By \eqref{eintersect}, we conclude by definition of $E_1$ that $\fc(\fp)\cap \fc(\fp')\neq \emptyset$. By +\eqref{eq freq dyadic} we conclude $\fc(\fp)\subset \fc(\fp')$. It follows that $\fp'\le \fp$. By maximality +\eqref{mnkmax} +of $\fp'$, we have $\fp'=\fp$. This proves the lemma. +\end{proof} + + +\begin{lemma}\label{adyadic} +For each $x\in A(\lambda,n,k)$, +there is a dyadic interval $I$ +that contains $x$ and is +a subset of +$I\subset A(\lambda,n,k)$. +\end{lemma} + +\begin{proof} + +Fix $k,n,\lambda,x$ as in the lemma. +Let $x\in A(\lambda,n,k)$. Let +$\mathcal{M}$ be the set of dyadic intervals + $\sc(\fp)$ +with $\fp$ in $\mathfrak{M}(n,k)$ +and $x\in \sc(\fp)$. By definition of +$A(\lambda,n,k)$, the cardinality of $\mathcal{M}$ +is at least $\lambda$. Let $I$ be an interval of +shortest length in $\mathcal{M}$. Then +$I$ is contained in all intervals of $\mathcal{M}$. +It follows that $I\subset A(\lambda,n,k)$. +\end{proof} + +\begin{lemma}\label{jn} + For all integers $k,n,\lambda\ge 0$, we have + \begin{equation}\label{alambdameasure} + \mu(A(\lambda,k,n)) \le 2^{k+1-\lambda}\mu(G)\, . + \end{equation} + + +\end{lemma} +\begin{proof} +Fix $k,n$ as in the lemma +and write short +$A(\lambda)$ for $A(\lambda,k,n)$. +We prove the lemma by induction on $\lambda$. +For $\lambda=0$, we use that $A(\lambda)$ by definition of $\mathfrak{M}(n,k)$ is contained in the union of elements in $ \mathcal{C}(G,k)$. Lemma \ref{ckmeasure} then completes the base of the induction. + +Now assume that the statement of Lemma \ref{jn} +is proven for some integer $\lambda\ge 0$. +The set $A(\lambda+1)$ is contained in the set $A(\lambda)$. +Let $\mathcal{M}$ be the set of dyadic intervals which are a subset of $A(\lambda)$. By Lemma \ref{adyadic}, the union of $\mathcal{M}$ is $A(\lambda)$. +Let $\mathcal{M}^*$ be the set of maximal intervals in $\mathcal{M}$. + +Let $L\in \mathcal{M}^*$. For each $x\in L$, we have +\begin{equation}\label{suminout} + \sum_{\fp \in {\mathfrak{M}}(n,k)} 1_{I(\fp)}(x)= + \sum_{\fp \in {\mathfrak{M}}(n,k):\sc(\fp) \subset L} 1_{I(\fp)}(x)+ + \sum_{\fp \in {\mathfrak{M}}(n,k):\sc(\fp) \not \subset L} 1_{I(\fp)}(x)\, . +\end{equation} +If the second sum on the right-hand-side is not zero, there is +an element of $\mathcal{D}$ containing $L$. +Let $\hat{L}$ be such interval with minimal $\sc(L)$. Then $\hat{L}$ is contained in $\sc(\fp)$ for all $\fp$ +contributing to the second sum in +\eqref{suminout}. +Hence the second sum in \eqref{suminout} is constant on +$\hat{L}$. +By maximality of $L$, the second sum is less than $1+\lambda 2^{n+1}$ somewhere on $\hat{L}$, and thus also +at $x$. +If $x$ is in addition in $A(\lambda+1)$, then +the left-hand-side of \eqref{suminout} is at least +$1+(\lambda+1) 2^{n+1}$, so we have by the triangle inequality for the first sum on the right-hand side +\begin{equation}\label{mnkonl} +\sum_{\fp \in {\mathfrak{M}}(n,k):\sc(\fp) \subset L} 1_{I(\fp)}(x)\ge 2^{n+1}\, .\end{equation} +By Lemma \ref{pairwise disjoint}, we have +\begin{equation} +\sum_{\fp \in {\mathfrak{M}}(n,k):\sc(\fp) \subset L} \mu({E_1}(\fp)) \leq \mu(L)\, . +\end{equation} +Multiplying by $2^n$ and applying \eqref{ebardense}, we obtain +\begin{equation}\label{mnkintl} + \sum_{\fp \in {\mathfrak{M}}(n,k):\sc(\fp) \subset L} \mu(\sc(\fp)) \leq 2^n \mu(L)\, . +\end{equation} +We then have with \eqref{mnkonl} and \eqref{mnkintl} +\begin{equation} +2^{n+1}\mu(A(\lambda+1)\cap L) = + \int_{A(\lambda+1)\cap L} 2^{n+1} d\mu +\end{equation} +\begin{equation} +\le + \int \sum_{\fp \in {\mathfrak{M}}(n,k):\sc(\fp) \subset L} 1_{I(\fp)} d\mu +\le 2^n \mu(L)\, . +\end{equation} +Hence +\begin{equation} + 2\mu(A(\lambda+1))=2\sum_{L\in \mathcal{M}^*} +\mu(A(\lambda+1)\cap L)\le +\sum_{L\in \mathcal{M}^*}\mu( L)= \mu(A(\lambda))\, . +\end{equation} +Using the induction hypothesis, this proves +\eqref{alambdameasure} for $\lambda+1$ and completes the proof of the lemma. +\end{proof} + +\begin{lemma}\label{g2bound} +We have +\begin{equation} + \mu(G_2)\le 2^{-4} \mu(G)\, . +\end{equation} +\end{lemma} +\begin{proof} + +We use Lemma \ref{jn} and sum twice a geometric series +to obtain +\begin{equation} + \sum_{0\le k}\sum_{k< n} +\mu(A(2n+6,k,n))\le \sum_{0\le k}\sum_{k< n} 2^{k-5-2n}\mu(G) +\end{equation} +\begin{equation} + \le \sum_{0\le k} 2^{-k-5}\mu(G)\le 2^{-4}\mu(G)\, . +\end{equation} +This proves the lemma. +\end{proof} + + +We turn to the set $G_3$. + +\begin{lemma}\label{musumlemma} + We have + \begin{equation}\label{eq musum} + \sum_{\mathfrak{m} \in \mathfrak{M}(n,k)} \mu(\sc(\mathfrak{m}))\le 2^{n+1}2^{k+1}\mu(G). + \end{equation} +\end{lemma} +\begin{proof} + We write the left-hand side of \ref{eq musum} +\begin{equation} + \int \sum_{\mathfrak{m} \in \mathfrak{M}(n,k)} 1_{\sc(\mathfrak{m})}(x) \, d\mu(x) \le +2^{n+1} \sum_{\lambda=1}^{|\mathfrak{M}|}\mu(A(\lambda, n,k))\,. +\end{equation} +Using Lemma \ref{alambdameasure} and then summing a geometric series, we estimate this by +\begin{equation} + \le +2^{n+1}\sum_{\lambda=1}^{|\mathfrak{M}|} +2^{k+1-\lambda}\mu(G) +\le +2^{n+1}2^{k+1}\mu(G)\, . +\end{equation} +This proves the lemma. +\end{proof} + + +\begin{lemma}\label{countu} +Let $n,k,j\ge 0$. We have for every $x\in X$ +\begin{equation} + \sum_{\fu\in \fU_1(n,k,j)} 1_{\sc(\fu)}(x) + \le 2^{1-j} + 2^{4a} \sum_{\mathfrak{m}\in \mathfrak{M}(n,k)} + 1_{\sc(\mathfrak{m})}(x) +\end{equation} +\end{lemma} + +\lars{Write consistently $Q$ or $\mathcal{Q}$ for the central frequency of a tile} + +\begin{proof} +Let $x\in X$. For each +$\fu\in \fU_1(n,k,j)$ with $x\in \sc(\fu)$, as $\fu \in \fC_1(n,k,j)$, +there are at least $2^{j-1}$ elements $\mathfrak{m}\in \mathfrak{M}(n,k)$ +with $2\fu \lesssim 100\mathfrak{m}$ and in particular +$x\in \sc(\mathfrak{m})$. Hence +\begin{equation}\label{ubymsum} + 1_{\sc(\fu)}(x) + \le 2^{1-j}\sum_{\mathfrak{m} \in \mathfrak{M}(n,k): 2\fu\lesssim 100 \mathfrak{m}} 1_{\sc(\mathfrak{m})}(x)\, . +\end{equation} +Conversely, for each $\mathfrak{m}\in \mathfrak{M}(n,k)$ +with $x\in \sc(\mathfrak{m})$, +let $\fU(\mathfrak{m})$ be the set of +$\fu\in \fU_1(n,k,j)$ with $x\in \sc(\fu)$ +and $2\fu \lesssim 100\mathfrak{m}$. +Summing \eqref{ubymsum} over $\fu$ and counting the pairs +$(\fu,\mathfrak{m})$ with $2\fu \lesssim 100\mathfrak{m}$ +differently gives +\begin{equation}\label{usumbymsum} + \sum_{\fu\in \fU_1(n,k,j)} 1_{\sc(\fu)}(x) + \le 2^{1-j}\sum_{\mathfrak{m} \in \mathfrak{M}(n,k)} + \sum_{\fu \in \fU(\mathfrak{m})} 1_{\sc(\mathfrak{m})}(x)\, . +\end{equation} + + + +We estimate the number of elements in $\fU(\mathfrak{m})$. +Let $\fu \in \fU(\mathfrak{m})$. +Then by definition of +$\fU(\mathfrak{m})$ +\begin{equation}\label{dby2} + d_{\fu}(\fcc(\fu),\fcc(\mathfrak{m}))\le 2\, . +\end{equation} +If $\fu'$ is a further element in $\fU(\mathfrak{m})$ with $\fu\neq \fu'$, then +\begin{equation} + \fcc(\mathfrak{m}) + \in B_{\fu}(\fcc(\fu),100)\cap B_{\fu'}(\fcc(\fu'),100)\ . +\end{equation} +By the last display and definition of $\fU_1(n,k,j)$, none of $\sc(\fu)$, $\sc(\fu')$ is strictly contained in the other. As both contain $x$, we have $\sc(\fu)=\sc(\fu')$. +We then have $d_{\fu}=d_{\fu'}$. + +By \eqref{eq freq comp ball}, the balls +$B_{\fu}(\fcc(\fu),0.2)$ and +$B_{\fu}(\fcc(\fu'),0.2)$ are +contained respectively in $\fc(\fu)$ +and $\fc(\fu')$ and thus are disjoint by \eqref{eq dis freq cover}. +By \eqref{dby2} and the triangle inequality, both balls are contained in $B_{\fu}(\fcc(\mathfrak{m}), 2.2)$. + +By \eqref{thirddb} applied four times, there is a collection of at most +$2^{4a}$ balls of radius $0.2$ with respect to the metric $d_{\fu}$ which cover the ball $B_{\fu}(\fcc(\mathfrak{m}),2.2)$. +Let $B'$ be a ball in this cover. +As the center of $B'$ can be in at most one of the disjoint balls +$B_{\fu}(\fcc(\fu),0.2)$ and +$B_{\fu}(\fcc(\fu'),0.2)$, +the ball $B'$ can contain at most +one of the points $\fu$, $\fu'$. + +Hence the set $\fU(\mathfrak{m})$ has at most +$2^{4a}$ many elements. +Inserting this into \eqref{usumbymsum} proves the lemma. +\end{proof} + +%For each $\fu\in \fU_1(n,k,j)$, define $\fL(\fu)$ +%to be the set of of all $\fp\in \fC_4(k,n,j)$ such that $2\fp\lesssim \fu$ +%and +%\begin{equation} +% B(c(\sc(\fp)), 8 D^{s(\fp)})\not \subset \sc(\fu)\, . +%\end{equation} + +\begin{lemma}\label{lulemma} +We have for each $\fu\in \fU_1(n,k,l)$, +\begin{equation} +\mu(\bigcup_{I\in \mathcal{L}(\fu)} I) +\le 2^{2a+2} D^{-\kappa Z(n+1)} + \mu(\sc(\mathfrak{u})). +\end{equation} + +\end{lemma} + +\lars{Write consistently $\ps$ or $s$.} + +\begin{proof} + Let $\fu\in \fU_1(n,k,l)$. +Let $I \in \mathcal{L}(\fu)$. Then we have $s(I) = s(\fu) - Z(n+1) - 1$ and $I \subset \sc(\fu)$ and $B(c(I), 8D^{s(I)}) \not \subset \sc(\fu)$. +By \eqref{eq vol sp cube}, the set $I$ +is contained in $B(c(I), 4D^{s(I)})$. +By the triangle inequality, the set $I$ +is contained in +\begin{equation} + X(\fu):=\{x \in \sc(\fu) \, : \, \rho(x, X \setminus \sc(\fu)) \leq 12 D^{s(\fu) - Z(n+1)-1}\}\,. +\end{equation} + By the small boundary property \eqref{eq small boundary}, we have + $$ + \mu(X(\fu)) \le + 2^{2a+2}(12 D^{-Z(n+1)-1})^\kappa + \mu(\sc(\mathfrak{u})). + $$ +Using $\kappa<1$ and $D \ge 12$, this proves the lemma. +\end{proof} + + + + + + + + + + + + + + + \begin{lemma}\label{g3bound} + + We have +\begin{equation} + \mu(G_3)\le 2^{-4} \mu(G)\, . +\end{equation} + \end{lemma} + + + + \begin{proof} +As each $\fp\in \fL_4(n,k,j)$ +is contained in $\cup\mathcal{L}(\fu)$ for some +$\fu\in \fU_1(n,k,l)$, we have +\begin{equation} +\mu(\bigcup_{\fp \in \fL_4 (n,k,j)}\sc(\fp)) +\le \sum_{\fu\in \fU_1(n,k,j)} +\mu(\bigcup_{I \in \mathcal{L} (\fu)}I). +\end{equation} +Using Lemma \ref{lulemma} and the Lemma \ref{countu}, we estimate this further + by +\begin{equation} + \le \sum_{\fu\in \fU_1(n,k,j)} + 2^{2a+2} D^{-\kappa Z(n+1)} + \mu(\sc(\mathfrak{u})) +\end{equation} +\begin{equation} + \le 2^{6a+3-j} \sum_{\mathfrak{m}\in \mathfrak{M}(n,k)} + D^{-\kappa Z(n+1)} + \mu(\sc(\mathfrak{m}))\,. +\end{equation} +%\begin{equation} +% \le 2^{1-j}\sum_{\mathfrak{m}\in \mathfrak{M}(n,k)} +% A^{30\ln_2A} D^{-\kappa Zn} +% \mu(\sc(\mathfrak{m}))\, , +%\end{equation} +Using Lemma \ref{musumlemma}, we estimate this by + \begin{equation} + \le +2^{6a + 3-j} D^{-\kappa Z(n+1)} + 2^{n+1}2^{k+1}\mu(G)\, . +\end{equation} +Now we estimate $G_3$ defined in \eqref{defineg3} by +\begin{equation} + \mu(G_3)\le \sum_{k\ge 0}\, \sum_{n \geq k}\, + \sum_{0\le j\le 2n+3} + \mu(\bigcup_{\fp \in \fL_4 (n,k,j)} + \sc(\fp)) +\end{equation} +\begin{equation} + \le \sum_{k\ge 0}\, \sum_{n \geq k}\, + \sum_{0\le j\le 2n+3} + 2^{6a + 5 + n + k -j} D^{-\kappa Z(n+1)}\mu(G) +\end{equation} +Summing geometric series, using $D^{\kappa Z}\ge 8$ \lars{assumption on $Z$}, we estimate this by +\begin{equation} + \le \sum_{k\ge 0}\, \sum_{n \geq k}\, + 2^{6a + 6 + n + k} D^{-\kappa Z(n+1)}\mu(G) +\end{equation} +\begin{equation} + = \sum_{k\ge 0} 2^{6a + 6 + 2k} D^{-\kappa Z(k+1)} \sum_{n \geq k}\, + 2^{n - k} D^{-\kappa Z(n-k)}\mu(G) +\end{equation} +\begin{equation} + \le \sum_{k\ge 0} 2^{6a + 7 + 2k} D^{-\kappa Z(k+1)}\mu(G) +\end{equation} +\begin{equation} + \le 2^{6a + 8} D^{-\kappa Z}\mu(G) +\end{equation} +Using $D = 2^{100a^2}$ and $a \ge 4$ and $\kappa Z \ge 1$ proves the lemma. +\end{proof} + +\section{Auxilliary lemmas} + +Before proving Lemma \ref{subsecflemma} and Lemma \ref{subsecalemma}, we collect some useful properties of $\lesssim$. + +\begin{lemma} + \label{lem wiggle monotone} + If $n\fp \lesssim m\fp'$ and + $n' \ge n$ and $m \ge m'$ then $n'\fp \lesssim m'\fp'$. +\end{lemma} + +\begin{proof} + This follows immediately from the definition \eqref{wiggleorder} of $\lesssim$ and the two inclusions $B_{\fp}(Q(\fp), n) \subset B_{\fp}(Q(\fp), n')$ and $B_{\fp'}(Q(\fp'), m') \subset B_{\fp'}(Q(\fp'), m)$. +\end{proof} + +\begin{lemma} + \label{lem aux wiggle} + Let $n, m \ge 1$. + If $\fp, \fp' \in \fP$ with $\sc(\fp) \ne \sc(\fp')$ and + \begin{equation} + \label{eq wiggle1} + n \fp \lesssim \fp' + \end{equation} + then + \begin{equation} + \label{eq wiggle2} + (n + 2^{-95 a} m) \fp \lesssim m\fp'\,. + \end{equation} +\end{lemma} + +\begin{proof} + The assumption \eqref{eq wiggle1} together with the definition \eqref{wiggleorder} of $\lesssim$ implies that $\sc(\fp) \subsetneq \sc(\fp')$. Let $q \in B_{\fp'}(Q(\fp'), 100)$. Then we have by the triangle inequality + $$ + d_{\fp}(Q(\fp), q) \le d_{\fp}(Q(\fp), Q(\fp')) + d_{\fp}(Q(\fp'), q) + $$ + Using \eqref{eq wiggle1} and \eqref{wiggleorder} for the first summand, and Lemma \ref{lem cube monotone} for the second summand, this is estimated by + $$ + n + 2^{-95a} d_{\fp'}(Q(\fp'), q) < n + 2^{-95a} m\,. + $$ + Thus $B_{\fp'}(Q(\fp'),n) \subset B_{\fp}(Q(\fp),n + 2^{-95a}m)$. Combined with $\sc(\fp) \subset \sc(\fp')$, this yields \eqref{eq wiggle2}. +\end{proof} + +\begin{lemma} + The following implications hold for all $\fq, \fq' \in \fP$: + \begin{equation} + \label{eq sc1} + \fq \lesssim \fq' \ \text{and} \ \lambda \ge 1 \implies \lambda \fq \lesssim \lambda \fq'\,, + \end{equation} + \begin{equation} + \label{eq sc2} + 10\fq \lesssim \fq' \ \text{and} \ \fq \ne \fq' \implies 100 \fq \lesssim 100 \fq'\,, + \end{equation} + \begin{equation} + \label{eq sc3} + 2\fq \lesssim \fq' \ \text{and} \ \fq \ne \fq' \implies 4 \fq \lesssim 500 \fq'\,. + \end{equation} +\end{lemma} + +\begin{proof} + All three implications are easy consequences of Lemma \ref{lem wiggle monotone}, Lemma \ref{lem aux wiggle} and the fact that $a \ge 4$. +\end{proof} + +\begin{lemma} + \label{lem rel claim} + If $\fu \sim \fu'$, then $\sc(u) = \sc(u')$ and $B_{\fu}(Q(\fu), 100) \cap B_{\fu'}(Q(\fu'), 100) \neq \emptyset$. +\end{lemma} + +\begin{proof} + Let $\fu, \fu' \in \fU_1(n,k,j)$ with $\fu \sim \fu'$. If $u = u'$ then the conclusion of the Lemma clearly holds. Else, there exists $\fp \in \fC_1(n,k,j)$ such that $\sc(\fp) \ne \sc(\fu)$ and $2 \fp \lesssim \fu$ and $10 \fp \lesssim \fu'$. + Using Lemma \ref{lem wiggle monotone} and \eqref{eq sc2}, we deduce that + \begin{equation} + \label{eq Fefferman trick0} + 100 \fp\lesssim 100 \fu\,, \qquad 100 \fp \lesssim 100\fu'\,. + \end{equation} + Now suppose that $B_{\fu}(Q(\fu), 100) \cap B_{\fu'}(Q(\fu'), 100) = \emptyset$. Then we have $\mathfrak{B}(\fu) \cap \mathfrak{B}(\fu') = \emptyset$, by the definition \eqref{defbfp} of $\mathfrak{B}$ and the definition \eqref{wiggleorder} of $\lesssim$, but also $\mathfrak{B}(\fu) \subset \mathfrak{B}(\fp)$ and $\mathfrak{B}(\fu') \subset \mathfrak{B}(\fp)$, by \eqref{defbfp}, \eqref{wiggleorder} and \eqref{eq Fefferman trick0}. + Hence, + $$ + |\mathfrak{B}(\fp)| \geq |\mathfrak{B}(\fu)| + |\mathfrak{B}(\fu')| \geq 2^{j} + 2^j = 2^{j+1}\,, + $$ + which contradicts $\fp \in \fC_1(n,k,j)$. Therefore we must have $B_{\fu}(Q(\fu), 100) \cap B_{\fu'}(Q(\fu'), 100) \ne \emptyset$. + + It follows from $2\fp \lesssim \fu$ and $10\fp \lesssim \fu'$ that $\sc(\fp) \subset \sc(\fu)$ and $\sc(\fp) \subset \sc(\fu')$. By \eqref{dyadicproperty}, it follows that $\sc(\fu)$ and $\sc(\fu')$ are nested. + Combining this with the conclusion of the last paragraph and definition \eqref{defunkj} of $\fU_1(n,k,j)$, we obtain that $\sc(\fu) = \sc(\fu')$. +\end{proof} + +We call a collection $\mathfrak{A}$ of tiles convex if +\begin{equation} + \label{eq convexity} + \fp \le \fp' \le \fp'' \ \text{and} \ \fp, \fp'' \in \mathfrak{A} \implies \fp' \in \mathfrak{A}\,. +\end{equation} + +\begin{lemma} + \label{lem convexity Pk} + For each $k$, the collection $\fP(k)$ is convex. +\end{lemma} + +\begin{proof} + Suppose that $\fp \le \fp' \le \fp''$ and $\fp, \fp'' \in \fP_k$. By \eqref{eq defPk} we have $\sc(\fp), \sc(\fp'') \in \mathcal{C}(G,k)$, so there exists by \eqref{muhj1} some $J \in \mathcal{D}$ with + $$ + \sc(\fp') \subset \sc(\fp'') \subset J + $$ + and $\mu(G \cap J) > 2^{-k-1} \mu(G)$. Thus \eqref{muhj1} holds for $\sc(\fp')$. On the other hand, by \eqref{muhj2}, there exists no $J \in \mathcal{D}$ with $\sc(\fp) \subset J$ and $\mu(G \cap J) > 2^{-k} \mu(G)$. Since $\sc(\fp) \subset \sc(\fp')$, this implies that \eqref{muhj2} holds for $\sc(\fp')$. Hence $\sc(\fp') \in \mathcal{C}(G,k)$, and therefore by \eqref{eq defPk} $\fp' \in \fP(k)$. +\end{proof} + +\begin{lemma} + \label{lem convexity Cnk} + For each $n,k$, the collection $\fC(n,k)$ is convex. +\end{lemma} + +\begin{proof} + Let $\fp \le \fp' \le \fp''$ with $\fp, \fp'' \in \fC(n,k)$. Then, in particular, $\fp, \fp'' \in \fP(k)$, so, by Lemma \ref{lem convexity Pk}, $\fp' \in \fP(k)$. Next, we show that if $\fq \le \fq' \in \fP(k)$ then $\dens'_k(\{\fq\}) \ge \dens_k'(\{\fq'\})$. If $\fp \in \fP(k)$ and $\lambda \ge 2$ with $\lambda \fq' \lesssim \lambda \fp$, then it follows from $\fq \le \fq'$, \eqref{eq sc1} and transitivity of $\lesssim$ that $\lambda \fq \lesssim \lambda \fp$. Thus the supremum in the definition \eqref{eq densdef} of $\dens_k'(\{\fq\})$ is over a superset of the set the supremum in the definition of $\dens_k'(\{\fq'\})$ is taken over, which shows $\dens'_k(\{\fq\}) \ge \dens_k'(\{\fq'\})$. From $\fp' \le \fp''$, $\fp'' \in \fC(n,k)$ and \eqref{def cnk} it then follows that + $$ + 2^{-n} < \dens_k'(\{\fp''\}) \le \dens_k'(\{\fp'\})\,. + $$ + Similarly, it follows from $\fp \le \fp'$, $\fp'' \in \fC(n,k)$ and \eqref{def cnk} that + $$ + \dens_k'(\{\fp'\}) \le \dens_k'(\{\fp\}) \le 2^{-n}\,. + $$ + Thus $\fp' \in \fC(n,k)$. +\end{proof} + +\begin{lemma} + \label{lem convexity C1} + For each $n,k,j$, the collection $\fC_1(n,k,j)$ is convex. +\end{lemma} + +\begin{proof} + Let $\fp\le\fp'\le\fp''$ with $\fp, \fp'' \in \fC_1(n,k,j)$. By Lemma \ref{lem convexity Cnk} and the inclusion $\fC_1(n,k,j) \subset \fC(n,k)$, which holds by definition \eqref{defcnkj}, we have $\fp' \in \fC(n,k)$. By \eqref{eq sc1} and transitivity of $\lesssim$ we have that $\fq \le \fq'$ and $2 \fq' \lesssim 100\mathfrak{m}$ imply $2\fq \lesssim 100\mathfrak{m}$. So, by \eqref{defbfp}, $\mathfrak{B}(\fp'') \subset \mathfrak{B}(\fp') \subset \mathfrak{B}(\fp)$. Consequently, by \eqref{defcnkj} + $$ + 2^j \le |\mathfrak{B}(\fp'')|\le |\mathfrak{B}(\fp')| \le |\mathfrak{B}(\fp)| < 2^{j+1}\,, + $$ + thus $\fp' \in \fC_1(n,k,j)$. +\end{proof} + +\begin{lemma} + \label{lem convexity C2} + For each $n,k,j$, the collection $\fC_2(n,k,j)$ is convex. +\end{lemma} + +\begin{proof} + Let $\fp \le \fp' \le \fp''$ with $\fp, \fp'' \in \fC_2(n,k,j)$. By \eqref{eq C2 def}, we have $\fC_2(n,k,j) \subset \fC_1(n,k,j)$. Combined with Lemma \ref{lem convexity C1}, it follows that $\fp' \in \fC_1(n,k,j)$. Suppose that $\fp' \notin \fC_2(n,k,j)$. By \eqref{eq C2 def}, this implies that there exists $0 \le l' \le Zn$ \lars{Z} with $\fp' \in \fL_1(n,k,j,l')$. By the definition \eqref{eq L1 def} of $\fL_1(n,k,j,l')$, this implies that $\fp$ is minimal with respect to $\le$ in $\fC_1(n,k,j) \setminus \bigcup_{l < l'} \fL_1(n,k,j,l)$. Since $\fp \in \fC_2(n,k,j)$ we must have $\fp \ne \fp'$. Thus $\fp \le \fp'$ and $\fp \ne \fp'$. By minimality of $\fp'$ it follows that $\fp \notin \fC_1(n,k,j) \setminus \bigcup_{l < l'} \fL_1(n,k,j,l)$. But by \eqref{eq C2 def} this implies $\fp \notin \fC_2(n,k,j)$, a contradiction. +\end{proof} + +\begin{lemma} + \label{lem convexity C3} + For each $n,k,j$, the collection $\fC_3(n,k,j)$ is convex. +\end{lemma} + +\begin{proof} + Let $\fp \le \fp' \le \fp''$ with $\fp, \fp'' \in \fC_3(n,k,j)$. By \eqref{eq C3 def} and Lemma \ref{lem convexity C2} it follows that $\fp' \in \fC_2(n,k,j)$. Suppose that $\fp' \notin \fC_3(n,k,j)$. Then, by \eqref{eq C3 def} and \eqref{eq L2 def}, there exists $\fu \in \fU_1(n,k,j)$ with $2\fp' \lesssim \fu$ and $\sc(\fp') \ne \sc(\fu)$. Together this gives $\sc(\fp') \subsetneq \sc(\fu)$. From $\fp' \le \fp$, \eqref{eq sc1} and transitivity of $\lesssim$ we then have $2\fp \lesssim \fu$. Also, $\sc(\fp) \subset \sc(\fp') \subsetneq \sc(\fu)$, so $\sc(\fp) \ne \sc(\fu)$. But then $\fp \in \fL_2(n,k,j)$, contradicting by \eqref{eq C3 def} the assumption $\fp \in \fC_3(n,k,j)$. +\end{proof} + +\begin{lemma} + \label{lem convexity C4} + For each $n,k,j$, the collection $\fC_4(n,k,j)$ is convex. +\end{lemma} + +\begin{proof} + Let $\fp \le \fp' \le\fp''$ with $\fp, \fp'' \in \fC_4(n,k,j)$. As before we obtain from the inclusion $\fC_4(n,k,j) \subset \fC_3(n,k,j)$ that $\fp' \in \fC_3(n,k,j)$. Thus, if $\fp' \notin \fC_4(n,k,j)$ then by \eqref{eq L3 def} there exists $l$ such that $\fp' \in \fL_3(n,k,j,l)$. Thus $\fp'$ is maximal with respect to $\le$ in $\fC_3(n,k,j) \setminus \bigcup_{0 \le l' < l} \fL_3(n,k,j,l')$. Since $\fp'' \in \fC_4(n,k,j)$ we must have $\fp' \ne \fp''$. Thus $\fp' \le \fp''$ and $\fp'\ne \fp''$. By minimality of $\fp'$ it follows that $\fp'' \notin \fC_3(n,k,j) \setminus \bigcup_{l < l'} \fL_3(n,k,j,l)$. But by \eqref{eq C4 def} this implies $\fp'' \notin \fC_4(n,k,j)$, a contradiction. +\end{proof} + +\begin{lemma} + \label{lem convexity C5} + For each $n,k,j$, the collection $\fC_5(n,k,j)$ is convex. +\end{lemma} + +\begin{proof} + Let $\fp \le \fp' \le\fp''$ with $\fp, \fp'' \in \fC_5(n,k,j)$. Then $\fp, \fp'' \in \fC_4(n,k,j)$ by \eqref{defc5}, and thus by Lemma \ref{lem convexity C4} also $\fp' \in \fC_4(n,k,j)$. Suppose that $\fp' \notin \fC_5(n,k,j)$. By \eqref{defc5}, it follows that $\fp' \in \fL_4(n,k,j)$. + By \eqref{eq L4 def}, there exists $\fu \in \fU_1(n,k,j)$ with $\sc(\fp') \subset \bigcup \mathcal{L}(\fu)$. Then also $\sc(\fp) \subset \bigcup \mathcal{L}(\fu)$, a contradiction. +\end{proof} + +\begin{lemma} + \label{lem dens comp} + We have for every $k\ge 0$ and $\fP'\subset \fP(k)$ +\begin{equation} + \dens_1(\fP')\le \dens_k'(\fP')\, . +\end{equation} +\end{lemma} +\begin{proof} +It suffices to show that for all $\fp'\in \fP'$ +and $\lambda>2$ and $\fp\in \fP(\fP')$ with $\lambda \fp' \lesssim \lambda \fp$ we have +\begin{equation} + \frac{\mu({E}_2(\lambda, \fp))}{\mu(\sc(\fp))} + \le \sup_{\fp'' \in \fP(k): \lambda \fp' \lesssim \lambda \fp''} + \frac{\mu({E}_2(\lambda, \fp''))}{\mu(\sc(\fp''))}. +\end{equation} + Let such $\fp'$, $\lambda$, $\fp$ be given. + It suffices to show that $\fp\in \fP(k)$, + that is, it satisfies \eqref{muhj1} + and \eqref{muhj2}. + +We show \eqref{muhj1}. + As $\fp\in \fP(\fP')$, there exists +$\fp''\in \fP'$ with $\sc(\fp')\subset \sc(\fp'')$. By assumption on $\fP'$, we have $\fp''\in \fP(k)$ and there exists +$J\in \mathcal{D}$ with + $\sc(\fp'')\subset J$ and + \begin{equation} + \mu(G\cap J)>2^{-k-1} \mu(J). + \end{equation} +Then also $\sc(\fp')\subset J$, which proves +\eqref{muhj1} for $\fp$. + +We show \eqref{muhj2}. Assume to get a contradiction that +there exists $J\in \mathcal{D}$ with + $\sc(\fp)\subset J$ and + \begin{equation}\label{mugj} + \mu(G\cap J)>2^{-k} \mu(J). + \end{equation} + As $\lambda\fp'\lesssim \lambda\fp$, we have $\sc(\fp')\subset \sc(\fp)$, and therefore + $\sc(\fp')\subset J$. This contradicts + $\fp'\in \fP'\subset \fP(k)$. This proves +\eqref{muhj2} for $\fp$. +\end{proof} + +\begin{lemma} + \label{lem 1density} + For each set $\mathfrak{A} \subset \mathfrak{C}(n,k)$, we have + $$ + \dens_1(\mathfrak{A}) \le 2^{4a}2^{-n+1}\,. + $$ +\end{lemma} + +\begin{proof} + We have by Lemma \ref{lem dens comp} that + $\dens_1(\mathfrak{A}) \le \dens_k'(\mathfrak{A})$. Since $\mathfrak{A} \subset \fC(n,k)$, it follows from monotonicity of suprema and the definition \eqref{eq densdef} that + $ + \dens_k'(\mathfrak{A}) \le \dens_k'(\fC(n,k))\,. + $ + By \eqref{eq densdef} and \eqref{def cnk}, we have + $$ + \dens_k'(\fC(n,k)) = \sup_{\fp \in \fC(n,k)} \dens_k'(\{\fp\}) \le 2^{4a}2^{-n+1}\,. + $$ +\end{proof} + +\section{Proof of Lemma \ref{subsecflemma}, the forests} +\label{subsecforest} + +Fix $k,n,j\ge 0$. +Define +$$ + \fC_6(n,k,j) +$$ +to be the set of all tiles $\fp \in \fC_5(n,k,j)$ such that $\sc(\fp) \not\subset G'$. The following chain of lemmata +establishes that the set $\fC_6(k,n,j)$ can be written as a union of a small number of $n$-forests. + +For $u\in \fU_1(n,k,j)$, define +\begin{equation} + \label{eq T1 def} + \mathfrak{T}_1(\fu):= \{\fp \in \fC_1(n,k,j) \ : \sc(\fp)\neq \sc(\fu), \ 2\fp \lesssim \fu\}\,. +\end{equation} +Define +\begin{equation} + \label{eq U2 def} + \fU_2(n,k,j) := \{ \fu \in \fU_1(n,k,j) \, : \, \mathfrak{T}_1(\fu) \cap \fC_6(n,k,j) \ne \emptyset\}\,. +\end{equation} +Define further a relation $\sim$ on $\fU_2(n,k,j)$ +by setting $\fu\sim \fu'$ +for $\fu,\fu'\in \fU_2(n,k,j)$ +if $\fu=\fu'$ or there exists $\fp$ in $\mathfrak{T}_1(\fu)$ +with $10 \fp\lesssim \fu'$. + +\begin{lemma} +\label{lem equivalence relation} +For each $n,k,j$, The relation $\sim$ on +$\fU_2(n,k,j)$ is an equivalence relation. +\end{lemma} + +\begin{proof} + Reflexivity holds by definition. + For transitivity, suppose that $\fu, \fu', \fu'' \in \fU_1(n,k,j)$ and $\fu \sim \fu'$, $\fu' \sim \fu''$. + By Lemma \eqref{lem rel claim}, it follows that $\sc(\fu) =\sc(\fu') = \sc(\fu'')$, that there exists $q_1 \in B_{\fu}(Q(\fu), 100) \cap B_{\fu'}(Q(\fu'), 100)$ and that there exists $q_2 \in B_{\fu'}(Q(\fu'), 100) \cap B_{\fu''}(Q(\fu''), 100)$. If $\fu = \fu'$, then $\fu \sim \fu''$ holds by assumption. Else, there exists by the definition of $\sim$ some $\fp \in \mathfrak{T}_1(\fu)$. + Then we have $2\fp \lesssim \fu$ and $\fp \ne \fu$ by definition of $\mathfrak{T}_1(\fu)$, so $4 \fp \lesssim 500 \fu$ by \eqref{eq sc3}. For $q \in B_{\fu''}(Q(\fu''), 1)$ it follows by the triangle inequality that + \begin{align*} + d_{\fu}(Q(\fu), q) &\le d_{\fu}(Q(\fu), q_1) + d_{\fu}(z_1, Q(\fu'))\\ + &\quad+ d_{\fu}(Q(\fu'), q_2) + d_{\fu}(z_2, Q(\fu'')) + + d_{\fu}(Q(\fu''), q)\,. + \end{align*} + Using \eqref{defdp} and the fact that $\sc(\fu) = \sc(\fu') = \sc(\fu'')$ this equals + \begin{align*} + &\quad d_{\fu}(Q(\fu), q_1) + d_{\fu'}(z_1, Q(\fu'))\\ + &\quad+ d_{\fu'}(Q(\fu'), q_2) + d_{\fu''}(z_2, Q(\fu'')) + + d_{\fu''}(Q(\fu''), q)\\ + &< 100 + 100 + 100 + 100 + 1 < 500\,. + \end{align*} + Since $4\fp \lesssim 500 \fu$, it follows that $d_{\fp}(Q(\fp), q) < 4 < 10$. We have shown that $B_{\fu''}(Q(\fu''), 1) \subset B_{\fp}(Q(\fp), 10)$, combining this with $\sc(\fu'') = \sc(\fu)$ gives $\fu \sim \fu''$. + + For symmetry suppose that $\fu \sim \fu'$. By Lemma \eqref{lem rel claim}, it follows that $\sc(\fu) = \sc(\fu')$ and that there exists $q_1 \in B_{\fu}(Q(\fu), 100) \cap B_{\fu'}(Q(\fu'), 100)$. Again, for $\fu = \fu'$ symmetry is obvious. If $\fu \ne \fu'$, then there exists $\fp \in \mathfrak{T}_1(\fu')$. By definition of $\mathfrak{T}_1(\fu')$, Lemma \ref{lem wiggle monotone} and \eqref{eq sc3}, it follows that + \begin{equation} + \label{eq rel1} + 10 \fq \lesssim 4\fp \lesssim 500 \fu'\,. + \end{equation} + If $q \in B_{\fu}(Q(\fu),1)$ then we have from the triangle inequality and the fact that $\sc(\fu) = \sc(\fu')$: + \begin{align*} + d_{\fu'}(Q(\fu'), q) &\le d_{\fu'}(Q(\fu'), q_1) + d_{\fu'}(q_1, Q(\fu)) + d_{\fu'}(Q(\fu), q)\\ + &= d_{\fu'}(Q(\fu'), q_1) + d_{\fu}(q_1, Q(\fu)) + d_{\fu}(Q(\fu), q)\\ + &< 100 + 100 + 1 < 500\,. + \end{align*} + Combing this with \eqref{eq rel1} and \eqref{wiggleorder}, we get $B_{\fu}(Q(\fu), 1) \subset B_{\fp}(Q(\fp), 10)$. Since $2\fp \lesssim \fu'$, we have $\sc(\fp) \subset \sc(\fu') = \sc(\fu)$. Thus, $10\fp \lesssim \fu$ which completes the proof of $\fu' \sim \fu$. +\end{proof} + +Choose a set $\fU_3(n,k,j)$ of representatives for the equivalence +classes of $\sim$ in $\fU_2(n,k,j)$. +Define for each $\fu\in \fU_3(n,k,j)$ +\begin{equation}\label{definesv} +\fT_2(\fu):= + \bigcup_{\fu\sim \fu'}\mathfrak{T}_1(\fu')\cap \fC_6(k,n,j)\, . +\end{equation} + +\begin{lemma} +\label{eq forest union} +We have +\begin{equation} + \fC_6(k,n,j)=\bigcup_{\fu\in \fU_3(n,k,j)}\mathfrak{T}_2(\fu)\, . +\end{equation} +\end{lemma} +\begin{proof} + Let $\fp \in \fC_6(n,k,j)$. + By \eqref{eq C4 def} and \eqref{defc5}, we have $\fp \in \fC_3(k,n,j)$. By \eqref{eq L2 def} and \eqref{eq C3 def}, there exists $\fu \in \fU_1(n,k,j)$ with $2\fp \lesssim \fu$ and $\sc(\fp) \ne \sc(\fu)$, that is, with $\fp \in \mathfrak{T}_1(\fu)$. Then $\mathfrak{T}_1(\fu)$ is clearly nonempty, so $\fu \in \fU_2(n,k,j)$. By the definition of $\fU_3(n,k,j)$, there exists $\fu' \in \fU_3(n,k,j)$ with $\fu \sim \fu'$. By \eqref{definesv}, we have $\fp \in \mathfrak{T}_2(\fu')$. +\end{proof} + +\begin{lemma} + \label{lem convex 6} + Let $\fu \in \fU_3(n,k,j)$. If $\fp \le \fp' \le \fp''$ and $\fp, \fp'' \in \mathfrak{T}_2(\fu)$, then $\fp' \in \mathfrak{T}_2(\fu)$. +\end{lemma} + +\begin{proof} + Suppose that $\fp, \fp'' \in \mathfrak{T}_2(\fu)$. By Lemma \eqref{lem convexity C5}, we have $\fp' \in \fC_5(n,k,j)$. Since $\fp \in \fC_6(n,k,j)$ we have $\sc(\fp) \not\subset G$, hence $\sc(\fp') \not \subset G$. This implies $\fp' \in \fC_6(n,k,j)$. Since $\fp'' \in \mathfrak{T}_2(\fu)$, we have $2\fp'' \lesssim \fu'$ and $\sc(\fp'')\ne\sc(\fu')$ for some $\fu' \sim \fp''$. By \eqref{eq sc1}, we have $2\fp' \lesssim 2\fp''$, so by transitivity of $\lesssim$ we have $2\fp' \lesssim \fu'$. Finally, $\sc(\fp') \subset \sc(\fp'')$ implies $\sc(\fp') \ne \sc(\fu')$, thus $\fp' \in \mathfrak{T}_1(\fu') \subset \mathfrak{T}_2(\fu)$. +\end{proof} + + +\begin{lemma} + \label{lem tree1 proof} + For each $\fu\in \fU_3(n,k,j)$, + the set $\mathfrak{T}_2(\fu)$ + satisfies \eqref{forest1}. +\end{lemma} +\begin{proof} + Let $\fp \in \mathfrak{T}_2(\fu')$. By \eqref{definesv}, there exists $\fu \sim \fu'$ with $\fp \in \mathfrak{T}_1(\fu)$. Then we have $2\fp \lesssim \fu$ and $\sc(\fp) \ne \sc(\fu)$, so by \eqref{eq sc3} $4\fp \lesssim 500\fu$. + Further, by Lemma \ref{lem rel claim}, we have that $\sc(\fu) = \sc(\fu')$ and there exists $q_1 \in B_{\fu}(Q(\fu),100) \cap B_{\fu'}(Q(\fu'),100)$. + Let $q \in B_{\fu'}(Q(\fu'), 1)$. + Using the triangle inequality and the fact that $\sc(\fu) =\sc(\fu')$, we obtain + \begin{align*} + d_{\fu}(Q(\fu), q) &\le d_{\fu'}(Q(\fu), q_1) + d_{\fu'}(Q(\fu'), q_1) + d_{\fu'}(Q(\fu'), q)\\ + &= d_{\fu}(Q(\fu), q_1) + d_{\fu'}(Q(\fu'), q_1) + d_{\fu'}(Q(\fu'), q)\\ + &< 100 + 100 + 1 < 500\,. + \end{align*} + Combining this with $4\fp \lesssim 500\fu$, we obtain + $$ + B_{\fu'}(Q(\fu'), 1) \subset B_{\fu}(Q(\fu), 500) \subset B_{\fp}(Q(\fp), 4)\,. + $$ + Together with $\sc(\fp) \subset \sc(\fu) = \sc(\fu')$, this gives $4\fp \lesssim \fu'$, which is \eqref{forest1}. +\end{proof} + +\begin{lemma} + \label{lem tree2 proof} + For each $\fu\in \fU_3(n,k,j)$, + the set $\mathfrak{T}_2(\fu)$ + satisfies the convexity condition \eqref{forest2}. +\end{lemma} + +\begin{proof} + Let $\fp, \fp'' \in \mathfrak{T}_2(\fu')$ and $\fp' \in \fP$ with $\fp \le \fp' \le \fp''$. By \eqref{definesv} we have $\fp, \fp'' \in \fC_6(n,k,j) \subset \fC_5(n,k,j)$. By Lemma \ref{lem convexity C5}, we have $\fp' \in \fC_5(n,k,j)$. Since $\fp \in \fC_6(n,k,j)$ we have $\sc(\fp) \not \subset G'$, so $\sc(\fp') \not \subset G'$ and therefore also $\fp' \in \fC_6(n,k,j)$. + + By \eqref{definesv} there exists $\fu \in \fU_1(n,k,j)$ with $\fp'' \in \mathfrak{T}_1(\fu)$ and hence $2\fp'' \lesssim \fu$ and $\sc(\fp'') \ne \sc(\fu)$. Together this implies $\sc(\fp'') \subsetneq \sc(\fu)$. With the inclusion $\sc(\fp') \subset \sc(\fp'')$ from $\fp' \le \fp''$, it follows that $\sc(\fp') \subsetneq \sc(\fu)$ and hence $\sc(\fp') \ne \sc(\fu)$. + By \eqref{eq sc1} and transitivity of $\lesssim$ we further have $2\fp' \lesssim \fu$, so $\fp' \in \mathfrak{T}_1(\fu)$. + It follows that $\fp' \in \mathfrak{T}_2(\fu')$, which shows \eqref{forest2}. +\end{proof} + +\begin{lemma} + \label{lem sep proof} + For each $\fu,\fu'\in \fU_3(n,k,j)$ with $\fu\neq \fu'$ and + and each $\fp \in \fT_2(\fu)$ + with $\sc(\fp)\subset \sc(\fu')$ we have + \ct{todo whats Z} + \begin{equation} + d_{\sc(\fp)}(\fcc(\fp), \fcc(\fu')) > 2^{Z(n+1)}\,. + \end{equation} +\end{lemma} + +\begin{proof} + By the definition \eqref{eq C2 def} of $\fC_2(n,k,j)$, there exists a tile $\fp' \in \fC_1(n,k,j)$ with $\fp' \le \fp$ and $s(\fp') = s(\fp)- Z(n+1)$. + By Lemma \ref{lem cube monotone} we have + $$ + d_{\sc(\fp)}(Q(\fp), Q(\fu')) \ge 2^{95a Z(n+1)} d_{\sc(\fp')}(Q(\fp), Q(\fu'))\,. + $$ + By \eqref{eq sc1} we have $2\fp' \lesssim 2\fp$, so by transitivity of $\lesssim$ there exists $\mathfrak{v} \sim \fu$ with $2\fp' \lesssim \mathfrak{v}$ and $\sc(\fp') \ne \sc(\mathfrak{v})$. Since $\fu, \fu'$ are not equivalent under $\sim$, we have $\mathfrak{v} \not \sim \fu'$, thus $10\fp' \not\lesssim \fu'$. This implies that there exists $q \in B_{\fu'}(Q(\fu'), 1) \setminus B_{\fp'}(Q(\fp'), 10)$. + + From $\fp' \le \fp$, $\sc(\fp') \subset \sc(\fp) \subset \sc(\fu')$ and Lemma \ref{lem cube monotone} it then follows that + \begin{align*} + &\quad d_{\sc(\fp')}(Q(\fp), Q(\fu'))\\ + &\ge -d_{\sc(\fp')}(Q(\fp), Q(\fp')) + d_{\sc(\fp')}(Q(\fp'), q) - d_{\sc(\fp')}(q, Q(\fu'))\\ + &\ge -d_{\sc(\fp')}(Q(\fp), Q(\fp')) + d_{\sc(\fp')}(Q(\fp'), q) - d_{\sc(\fu')}(q, Q(\fu'))\\ + &> -1 + 10 - 1 = 8\,. + \end{align*} + The Lemma follows by combining the two displays with the fact that $95 a \ge 1$. +\end{proof} + +\begin{lemma} + \label{lem normal proof} + For each $\fu\in \fU_3(n,k,j)$ + and each $\fp \in \mathfrak{T}_2(\fu)$ + we have + \begin{equation} + B(c(\sc(\fp)), 8 D^{s(\fp)}) \subset \sc(\fu). + \end{equation} +\end{lemma} + +\begin{proof} + Let $\fp \in \mathfrak{T}_2(\fu)$. Let $\fq$ be a maximal element with respect to $\le$ in + $$ + \bigcup_{\fu \sim \fu'} \mathfrak{T}_1(\fu') \cap \fC_3(n,k,j)\,. + $$ + We now show that there is no $\fq' \in \fC_3(n,k,j)$ with $\fq \le \fq'$ and $\fq \ne \fq'$. Indeed, suppose $\fq'$ was such a tile. Then $2\fq' \not \lesssim \fu'$ for any $\fu' \sim \fu$, by maximality of $\fq$. But by \eqref{eq C3 def} there exists $\fu'' \in \fU_1(n,k,j)$ with $2\fq' \lesssim \fu''$. By Lemma \ref{lem wiggle monotone}, this implies $\fu \sim \fu''$, so $\fq' \in \mathfrak{T}_2(\fu)$, a contradiction. + + By Lemma \ref{lem rel claim}, we have $s(\fq) < s(\fu)$. By definition of $\fC_4(n,k,j)$, $\fp$ is not in any of the maximal $Z(n+1)$ layers of tiles in $\fC_3(n,k,j)$, and hence $s(\fp) \le s(\fq) - Z(n+1) \le s(\fu) - Z(n+1) - 1$. + + Thus, there exists some cube $I \in \mathcal{D}$ with $s(I) = s(\fu) - Z(n+1) - 1$ and $I \subset \sc(\fu)$ and $\sc(\fp) \subset I$. Since $\fp \in \fC_5(n,k,j)$, we have that $I \notin \mathcal{L}(\fu)$, so $B(c(I), 8D^{s(I)}) \subset \sc(\fu)$. By the triangle inequality, \eqref{defineD} and $a \ge 4$, the same then holds for the subcube $\sc(\fp) \subset I$. +\end{proof} + + +\begin{lemma} + \label{lem overlap} + It holds that + \begin{equation} + \sum_{\fu \in \fU_3(n,k,j)} \mathbf{1}_{\sc(\fu)} \le 1 + (4n+12)2^{n}\,. + \end{equation} +\end{lemma} + +\begin{proof} + Suppose that a point $x$ is contained in more than $1 + (4n + 12)2^n$ cubes $\sc(\fu)$ with $\fu \in \fU_3(n,k,j)$. Since $\fU_3(n,k,j) \subset \fC_1(n,k,j)$ For each such $\fu$, there exist $\mathfrak{m}(\fu) \in \mathfrak{M}(n,k)$ with $\fu \le \mathfrak{m}(\fu)$. The map $\fu \mapsto\mathfrak{m}(\fu)$ is injective: If $\fu \le \mathfrak{m}$ and $\fu' \le \mathfrak{m}$, then $\sc(\fu) \subset \sc(\fu')$ or $\sc(\fu') \subset \sc(\fu)$ by \eqref{dyadicproperty}. Hence, by \eqref{defunkj}, $B_{\fu}(Q(\fu),100) \cap B_{\fu'}(Q(\fu'), 100) = \emptyset$. This contradicts $\Omega(\mathfrak{m})$ being contained in both sets. Thus $x$ is contained in at least $1 + (4n + 12)2^n$ cubes $\sc(\mathfrak{m})$, $\mathfrak{m} \in \mathfrak{M}(n,k)$. Consequently, we have by \eqref{eq Aoverlap def} that $x \in A(2n + 6, n,k) \subset G_2$. Let $\sc(\fu)$ be an inclusion minimal cube among the $\sc(\fu'), \fu' \in \fU_3(n,k,j)$ with $x \in \sc(\fu)$. By the dyadic property \eqref{dyadicproperty}, we have $\sc(\fu) \subset \sc(\fu')$ for all cubes $\sc(\fu')$ containing $x$. Thus + $$ + \sc(\fu) \subset \{y \ : \ \sum_{\fu \in \fU_3(n,k,j)} \mathbf{1}_{\sc(\fu)}(y) > 1 + (4n+12)2^{n}\} \subset G_2\,. + $$ + Thus $\mathfrak{T}_1(n,k,j) \cap \fC_6(n,k,j) = \emptyset$. + This contradicts $\fu \in \fU_2(n,k,j)$. +\end{proof} + +\begin{proof}[Proof of Lemma \ref{subsecflemma}] + We first fix $n, k, j$. + By \eqref{definetp} and \eqref{definedE}, we have that + $\mathbf{1}_{\sc(\fp)} T_{\fp}f(x) = T_{\fp}f(x)$ and hence $\overline{g(x)} T_{\fp}f(x)= 0$ for all $\fp \in \fC_5(n,k,j) \setminus \fC_6(n,k,j)$. + Thus it suffices to estimate the contribution of the sets $\fC_6(n,k,j)$. By Lemma \ref{lem overlap}, we can decompose $\fU_3(n,k,j)$ as a disjoint union of at most $4n + 13$ collections $\fU_4(n,k,j,l)$, $1 \le l \le 4n+13$, each satisfying + $$ + \sum_{\fu \in \fU_4(n,k,j,l} \mathbf{1}_{\sc(\fu)} \le 2^n\,. + $$ + By Lemmas \ref{lem tree1 proof}, \ref{lem tree2 proof}, \ref{lem sep proof}, \ref{lem normal proof} and \ref{lem 1density}, the pairs + $$ + (\fU_4(n,k,j,l), \mathfrak{T}_2|_{\fU_4(n,k,j,l)}) + $$ + are $n$-forests for each $n,k,l$, and by Lemma \ref{eq forest union}, we have + $$ + \fC_6(n,k,j) = \bigcup_{l = 1}^{4n + 13} \bigcup_{\fu \in \fU_4(n,k,j,l)} \mathfrak{T}_2(\fu)\,. + $$ + Since $\sc(\fp) \not\subset G_1$ for all $\fp \in \fC_6(n,k,j)$, we have $\fC_6(n,k,j) \cap \fP_{F,G} = \emptyset$ and hence + $$ + \dens_2(\bigcup_{\fu \in \fU_4(n,k,j,l)} \mathfrak{T}_2(\fu)) \le 2^{2a + 5} \frac{\mu(F)}{\mu(G)}\,. + $$ + Using the triangle inequality according to the splitting by $k,n,j$ and $l$ in \eqref{disclesssim1}, applying Proposition \ref{forestprop} to each term and summing the resulting geometric series, we obtain Lemma \ref{subsecflemma}. +\end{proof} + +\section{Proof of Lemma \ref{subsecalemma}, the antichains} +\label{subsecantichain} + +Define $\fP_{X \setminus G'}$ to be the set of all $\fp \in \fP$ such that $\sc(\fp) \not \subset G'$. +\begin{lemma} + We have that + \begin{align} + \label{eq fp' decomposition} + &\quad \fP' \cap \fP_{X \setminus G'}\\ + &= \bigcup_{k \ge 0} \bigcup_{n \ge k} \fL_0(n,k) \cap \fP_{X \setminus G'} \\ + &\quad\cup \bigcup_{k \ge 0} \bigcup_{n \ge k}\bigcup_{0 \le j \le 2n+3} \fL_2(n,k,j) \cap \fP_{X \setminus G'}\\ + &\quad\cup \bigcup_{k \ge 0} \bigcup_{n \ge k}\bigcup_{0 \le j \le 2n+3} \bigcup_{0 \le l \le Z(n+1)} \fL_1(n,k,j,l) \cap \fP_{X \setminus G'}\\ + &\quad\cup \bigcup_{k \ge 0} \bigcup_{n \ge k}\bigcup_{0 \le j \le 2n+3} \bigcup_{0 \le l \le Z(n+1)} \fL_3(n,k,j,l)\cap \fP_{X \setminus G'}\,. + \end{align} +\end{lemma} + +\begin{proof} + Let $\fp \in \fP' \cap \fP_{X \setminus G'}$. Clearly, for every cube $J \in \mathcal{D}$ there exists some $k \ge 0$ such that \eqref{muhj1} holds, and for no cube $J \in \mathcal{D}$ and no $k < 0$ does \eqref{muhj2} hold. Thus $\fp \in \fP(k)$ for some $k \ge 0$. + + Next, since $E_2(\lambda, \fp') \subset \sc(\fp')\cap G$ for every $\lambda \ge 2$ and every tile $\fp' \in \fP(k)$ with $\lambda\fp \lesssim \lambda \fp'$, it follows from \eqref{muhj2} that $\mu(E_2(\lambda, \fp')) \le 2^{-k} \mu(\sc(\fp'))$ for every such $\fp'$, so $\dens_k'(\{\fp\}) \le 2^{-k}$. Combining this with $a \ge 0$, it follows from \eqref{def cnk} that there exists $n\ge k$ with $\fp \in \fC(n,k)$. + + Since $\fp \not \in \fP_{X \setminus G'}$, we have in particular $\fp \not \subset A(2n + 6, k, n)$, so there exist at most $1 + (4n + 12)2^n < 2^{2n+4}$ tiles $\mathfrak{m} \in \mathfrak{M}(n,k)$ with $\fp \le \mathfrak{m}$. It follows that $\fp \in \fL_0(n,k)$ or $\fp \in \fC_1(n,k,j)$ for some $1 \le j \le 2n + 3$. In the former case we are done, in the latter case the inclusion to be shown follows immediately from the definitions of the collections $\fC_i$ and $\fL_i$. +\end{proof} + +\begin{lemma} + We have that + $$ + \fL_0(n,k) = \dot{\bigcup_{1 \le l \le n}} \fL_0(n,k,l)\,, + $$ + where each $\fL_0(n,k,l)$ is an antichain. +\end{lemma} + +\begin{proof} + It suffices to show that $\fL_0(n,k)$ contains no chain of length $n + 1$. Suppose that we had such a chain $\fp_0 \le \fp_1 \le \dotsb \le \fp_{n}$ with $\fp_i \ne \fp_{i+1}$ for $i =0, \dotsc, n-1$. By \eqref{def cnk}, we have that $\dens_k'(\{\fp_n\}) > 2^{-n}$. Thus, by \eqref{eq densdef}, there exists $\fp' \in \fP(k)$ and $\lambda \ge 2$ with $\lambda \fp_n \le \lambda \fp'$ and + \begin{equation} + \label{eq p'} + \frac{\mu(E_2(\lambda, \fp'))}{\mu(\sc(\fp'))} > \lambda^{a} 2^{4a} 2^{-n}\,. + \end{equation} + Let $\mathfrak{O}$ be the set of all $\fp'' \in \fP_k$ such that we have $ \sc(\fp'') = \sc(\fp')$ and $B_{\fp'}(Q(\fp'), \lambda) \cap \Omega(\fp'') \neq \emptyset$. + We now show that + \begin{equation} + \label{eq O bound} + |\mathfrak{O}| \le 2^{4a}\lambda^a\,. + \end{equation} + The balls $B_{\fp'}(Q(\fp''), 0.2)$, $\fp'' \in \mathfrak{O}$ are disjoint by \eqref{eq freq comp ball}, and by the triangle inequality contained in $B_{\fp'}(Q(\fp'), \lambda+1)$. By assumption the \eqref{thirddb} on $\Theta$, this ball can be covered with + $$ + 2^{a(\lceil \log_2(\lambda+1)\rceil + 2)} \le 2^{a(\log_2(\lambda) + 4)} = 2^{4a}\lambda^a + $$ many $d_{\fp'}$-balls of radius $1/4$. By the triangle inequality, each such ball contains at most one $Q(\fp'')$, and each $Q(\fp'')$ is contained in one of the balls. Thus we get \eqref{eq O bound}. + + By \eqref{definee1} and \eqref{definee2} we have $E_2(\lambda, \fp') \subset \bigcup_{\fp'' \in \mathfrak{O}} E_1(\fp'')$, thus + $$ + 2^{4a}\lambda^a 2^{-n} < \sum_{\fp'' \in \mathfrak{O}} \frac{\mu(E_1(\fp''))}{\mu(\sc(\fp''))}\,. + $$ + Hence there exists a tile $\fp'' \in \mathfrak{O}$ with + \begin{equation*} + \mu(E_1(\fp'')) \ge 2^{-n} \mu(\sc(\fp'))\,. + \end{equation*} + By the definition \eqref{mnkmax} of $\mathfrak{M}(n,k)$, there exists a tile $\mathfrak{m} \in \mathfrak{M}(n,k)$ with $\fp' \leq \mathfrak{m}$. From \eqref{eq p'}, the inclusion $E_2(\lambda, \fp') \subset \sc(\fp')$ and $a\ge 1$ we obtain + $$ + 2^n \geq 2^{4a} \lambda^{a} \geq \lambda\,. + $$ + From the triangle inequality, Lemma \ref{lem cube monotone} and $a \ge 1$, we now obtain for all $q \in B_{\mathfrak{m}}(Q(\mathfrak{m}), 100)$ that + \begin{align*} + d_{\fp_0}(Q(\fp_0), q) + &\leq d_{\fp_0}(Q(\fp_0), Q(\fp_{n})) + d_{\fp_0}(Q(\fp_{n}), Q(\fp')) + d_{\fp_0}(Q(\fp'), Q(\fp''))\\ + &\quad+ d_{\fp_0}(Q(\fp''), Q(\mathfrak{m})) + + d_{\fp_0}(Q(\mathfrak{m}), Q)\\ + &\leq 1 + 2^{-95an} (d_{\fp_{n}}(Q(\fp_n), Q(\fp')) + d_{\fp'}(Q(\fp'), Q(\fp''))\\ + &\quad+ d_{\fp''}(Q(\fp''), Q(\mathfrak{m})) + + d_{\mathfrak{m}}(Q(\mathfrak{m}), q))\\ + &\leq 1 + 2^{-95an}(\lambda + (\lambda + 1) + 1 + 100) \leq 2\,. + \end{align*} + Thus, by \eqref{straightorder}, $2\fp_0 \leq 100\mathfrak{m}$, a contradiction to $\fp_0 \notin \fC(n,k)$. +\end{proof} + +\begin{lemma} + Each of the sets $\fL_2(n,k,j)$ is an antichain. +\end{lemma} + +\begin{proof} + Suppose that there are $\fp_0, \fp_1 \in \fL_2(n,k,j)$ with $\fp_0 \ne \fp_1$ and $\fp_0 \le \fp_1$. By Lemma \ref{lem wiggle monotone} and Lemma \ref{lem aux wiggle}, it follows that $2\fp_0 \lesssim 200\fp_1$. Since $\fL_2(n,k,j)$ is finite, there exists a maximal $l \ge 1$ such that there exists a chain $2\fp_0 \lesssim 200 \fp_1 \lesssim \dotsb \lesssim 200 \fp_l$ with $\fp_i \ne \fp_{i+1}$ for $i = 0, \dotsc, l-1$. + If we have $\fp_l \in \fU_1(n,k,j)$, then it follows from $2\fp \lesssim 200 \fp_l \lesssim \fp_l$ and \eqref{eq L2 def} that $\fp \not\in \fL_2(n,k,j)$, a contradiction. Thus, by the definition \eqref{defunkj} of $\fU_1(n,k,j)$, there exists $\fp_{l+1} \in \fC_1(n,k,j)$ with $\sc(\fp_l) \subsetneq \sc(\fp_{l+1}) $ and $q \in B_{\fp_l}(Q(\fp_l), 100) \cap B_{\fp_{l+1}}(Q(\fp_{l+1}), 100)$. Using the triangle inequality and Lemma \ref{lem cube monotone}, one deduces that $200 \fp_l \lesssim 200\fp_{l+1}$. This contradicts maximality of $l$. +\end{proof} + +\begin{lemma} + Each of the sets $\fL_1(n,k,j,l)$ and $\fL_3(n,k,j,l)$ is an antichain. +\end{lemma} + +\begin{proof} + By its definition \eqref{eq L1 def}, each set $\fL_1(n,k,j,l)$ is a set of minimal elements in some set of tiles with respect to $\le$. If there were distinct $\fp, \fq \in \fL_1(n,k,j,l)$ with $\fp \le \fq$, then $\fq$ would not be minimal. Hence such $\fp, \fq$ do not exist. Similarly, be \eqref{eq L3 def}, each set $\fL_3(n,k,j,l)$ is a set of maximal elements in some set of tiles with respect to $\le$. If there were distinct $\fp, \fq \in \fL_3(n,k,j,l)$ with $\fp \le \fq$, then $\fp$ would not be maximal. +\end{proof} + +\begin{proof}[Proof of Lemma \ref{subsecalemma}] + If $\fp \not\in \fP_{X \setminus G'}$, then $\sc(\fp) \subset G'$. By \eqref{definetp} and \eqref{definee1}, it follows that + $\mathbf{1}_{G \setminus G'} T_{\fp}f(x) = 0$. We thus have + $$ + \overline{g(x)} \sum_{\fp \in \fP'} T_{\fp}f(x) = \overline{g(x)} \sum_{\fp \in \fP' \cap \fP_{X \setminus G'}} T_{\fp}f(x)\,. + $$ + Let $\fL(n,k)$ denotes any of the terms $\fL_i(n,k,j,l) \cap \fP_{X \setminus \fP'}$ on the right hand side of \eqref{eq fp' decomposition}, where the indices $j, l$ may be void. Then $\fL(n,k)$ is an antichain, by the three preceeding Lemmas. Further, we have $\dens_1(\fL(n,k)) \le 2^{4a+1 - n}$ by Lemma \ref{lem 1density}, and $\dens_2(\fL(n,k)) \le 2^{2a+5} \frac{\mu(F)}{\mu(G)}$ since $\fL(n,k) \cap \fP_{F,G} \subset \fP_{X \setminus \fP'} \cap \fP_{F, G} = \emptyset$. + Applying now the triangle inequality according to the decomposition \eqref{eq fp' decomposition}, and then applying Proposition \ref{antichainprop} to each term, interchanging the $n$ and $k$ summation and then summing the resulting geometric series, we obtain the claimed estimate. \lars{Possibly write down the computation} +\end{proof} + +\chapter{Proof of Proposition \ref{antichainprop}} + +\label{antichainboundary} + +Let an antichain $\mathfrak{A}$ +and functions $f$, $g$ as in Proposition \ref{antichainprop} be given. +We prove \eqref{eq antiprop} +in Subsection \ref{sec TT* T*T} +as the geometric mean of two inequalities, +each involving one of the two densities. +One of these two inequalities will need a careful estimate formulated in +Lemma \ref{lem basic TT*} of +the $TT^*$ correlation between two tile operators of two tiles. +Lemma \ref{lem basic TT*} will be proven in +Subsection \ref{sec tile operator} +The summation of the contributions of these individual correlations will require a +geometric Lemma \ref{lem antichain 1} counting the relevant tile pairs. +Lemma \ref{lem antichain 1} will be proven in Subsection +\ref{subsec geolem}. + + + + + + +\section{The density arguments}\label{sec TT* T*T} + +We begin with the following crucial disjointness property of the sets $E(\fp)$ with $\fp \in \mathfrak{A}$. +\begin{lemma} +\label{lem antichain -1} +Let $\fp,\fp'\in \mathfrak{A}$. +If there exists an $x\in X$ with $x\in E(\fp)\cap E(\fp')$, +then $\fp= \fp'$. +. +\end{lemma} +\begin{proof} +Let $\fp,\fp'$ and $x$ be given. +Assume without loss of generality that $\ps(\fp)\le \ps(\fp')$. +As we have $x\in E(\fp)\subset \sc(\fp)$ and $x\in E(\fp')\subset \sc(\fp')$ by Definition \eqref{defineep}, we conclude +withfor $i=1,2$ that +$\tQ(x)\in\fc(\fp)$ and $\tQ(x)\in\fc(\fp')$. By \eqref{eq freq dyadic} we have $\fc(\fp')\subset \fc(\fp)$. By Definition +\eqref{straightorder}, we conclude $\fp\le \fp'$. As $\mathfrak{A}$ is an antichain, we conclude $\fp=\fp'$. +This proves the lemma. +\end{proof} + + + +Let $\mathcal{B}$ be the collection of balls +\begin{equation} + B(\pc(\fp), 8D^{\ps(\fp)})\, . +\end{equation} +with $\fp\in \mathfrak{A}$ and recall the definition of +$M_{\mathcal{B}}$ from Proposition \ref{prop hlm}. +\begin{lemma}\label{lem hlmbound} +Let $x\in X$. +Then +\begin{equation}\label{hlmbound} + | \sum_{\fp \in \mathfrak{A}}T_{\fp} f(x)|\le 2^{107 a^3} M_{\mathcal{B}} f (x) \, . +\end{equation} +\end{lemma} + + + +\begin{proof} +Fix $x\in X$. By Lemma \ref{lem antichain -1}, there is at most one $\fp \in \mathfrak{A}$ +such that + $T_{\fp} f(x)$ is not zero. + If there is no such $\fp$, the estimate \eqref{hlmbound} follows. + + Assume there is such a $\fp$. + By definition of $T_{\fp}$ we have $x\in E(\fp)\subset \sc(\fp)$ and by the squeezing property \eqref{eq vol sp cube} +\begin{equation}\label{eqtttt0} + \rho(x, \pc(\fp))\le 4D^{\ps(\fp)}\, . +\end{equation} + +Let $y\in X$ with $K_{\ps(\fp)}(x,y)\neq 0$. By Definition \eqref{defks} of $K_{\ps(\fp)}$ +we have +\begin{equation}\label{supp Ks1} + \frac{1}{4} D^{\ps(\fp)-1} + \leq \rho(x,y) \leq \frac{1}{2} D^{\ps(\fp)}\, . +\end{equation} +As by the squeezing property \eqref{eq vol sp cube}, we have +\begin{equation} + \rho(\pc(\fp),x)\le 4D^{\ps(\fp)} +\end{equation} +The triangle inequality with \eqref{eqtttt0} an \eqref{supp Ks1} implies +\begin{equation} + \rho(\pc(\fp),y)\le 8D^{\ps(\fp)}\, . +\end{equation} +Using the kernel bound \eqref{eqkernel size} and the lower bound in \eqref{supp Ks} +we obtain +\begin{equation} +|K_{\ps(\fp)}(x,y)|\le \frac{2^{a^3}}{\mu(B(x,\frac 14 D^{{\ps(\fp)}-1}))}\, . +\end{equation} +Using $D=2^{100a^2}$ +and the doubling property \eqref{doublingx} $5 +100a^2$ times estimates +the last display by +\begin{equation} +\le \frac{2^{5a+101a^3}}{\mu(B(x, 8D^{\ps(\fp)}))}\, . +\end{equation} + Using that $|\mfa|$ is bounded by $1$ + for every $\mfa\in \Mf$, we estimate with the triangle inequality and the above information + \begin{equation} + | T_{\fp} f(x)| + \le \frac{2^{5a+101 a^3}}{\mu(B(x, 8D^{\ps(\fp)}))} \int _{\mu(B(x, 8D^{\ps(\fp)}))} |f(y)|\, dy \end{equation} +This together with $a\ge 1$ proves the Lemma. +\end{proof} + +Set +\begin{equation} + \tilde{q}=\frac {2q}{1+q} +\end{equation} +and note that with $1< q\le 2$ we conclude $1<\tilde{q}4$ and thus $14$. + + + + +\end{proof} + + +We have +\begin{equation} + \left (\frac 1{\tilde{q}} -\frac 12\right) (2-q)= \frac 1q -\frac 12 +\end{equation} +Multiplying the $(2-q)$-th power of \eqref{eqttt9} and the $(q-1)$-th power of \eqref{eqttt3} +and estimating gives after simplification of some factors gives +\begin{equation}\label{eqttt8} + |\int \overline{g(x)} \sum_{\fp \in \mathfrak{A}} T_{\fp} f(x)\, d\mu(x)| + \end{equation} + \begin{equation} + \le 2^{200a^3}({q}-1)^{-1} \tau^{-1}\dens_1(\mathfrak{A})^{\frac {q-1}{2p}}\dens_2(\mathfrak{A})^{\frac 1{q}-\frac 12} \|f\|_2\|g\|_2\, . + \end{equation} + With the definiiton of $p$, this implies +Proposition \ref{antichainprop}. + + +\section{Proof of L. \ref{lem basic TT*}, the tile correlation bound }\label{sec tile operator} +We begin with the following basic estimates for $K_s$. +\begin{lemma} +Let $-S\le s\le S$ and $x,y,y'\in X$. +If $K_s(x,y)\neq 0$, then we have +\begin{equation}\label{supp Ks} + \frac{1}{4} D^{s-1} \leq \rho(x,y) \leq \frac{1}{2} D^s\, . +\end{equation} +We have +\begin{equation} + \label{eq Ks size} + |K_s(x,y)|\le \frac{2^{102 a^3}}{\mu(B(x, D^{s}))}\, +\end{equation} +and \begin{equation} + \label{eq Ks smooth} + |K_s(x,y)-K_s(x, y')|\le \frac{2^{150a^3}}{\mu(B(x, D^{s}))} + \left(\frac{ \rho(y,y')}{D^s}\right)^{\tau}\,. +\end{equation} +\end{lemma} + +\begin{proof} +By Definition \eqref{defks}, the function $K_s$ is the product of +$K$ with a function which is supported in the set of all +$x,y$ satisfying \eqref{supp Ks}. This proves \eqref{supp Ks}. + +Using \eqref{eqkernel size} and the lower bound in \eqref{supp Ks} +we obtain +\begin{equation} +|K_s(x,y)|\le \frac{2^{a^3}}{\mu(B(x,\frac 14 D^{s-1}))} +\end{equation} +Using $D=2^{100a^2}$ +and the doubling property \eqref{doublingx} $2 +100a^2$ times estimates +the last display by +\begin{equation} +\le \frac{2^{2a+101a^3}}{\mu(B(x, D^{s}))}\, . +\end{equation} +Using $a\ge 4$ proves \eqref{eq Ks size}. + + +Similarly, we obtain with \eqref{eqkernel y smooth} and the lower bound in +\eqref{supp Ks} +\begin{equation} + |K_s(x,y)-K_s(x, y')|\le \frac{2^{a^3}}{\mu(B(x, \frac 14 D^{s-1}))} + \left(\frac{ \rho(y,y')}{\frac 14 D^{s-1}}\right)^{\tau}\,. +\end{equation} +Using $\tau\le 1$, this is estimated by +\begin{equation} + \le \frac{4D 2^{2a+101a^3}}{\mu(B(x, D^{s}))} + \left(\frac{ \rho(y,y')}{D^{s}}\right)^{\tau} + = \frac{2^{2+2a+100a^2+101a^3}}{\mu(B(x, D^{s}))} + \left(\frac{ \rho(y,y')}{D^{s}}\right)^{\tau}\,. +\end{equation} +Using $a\ge 4$, this proves \eqref{eq Ks smooth}. +\end{proof} +The next Lemma prepares an application of +Proposition \ref{lem vdc regularity}. +\begin{lemma}\label{lem ksquare} +Let $-S\le s_1\le s_2\le S$ and let $x_1,x_2\in X$. +Define \begin{equation} + \varphi(y) := \overline{K_{s_1}(x_1, y)} + K_{s_2}(x_2, y) \, . +\end{equation} +If $\varphi(y)\neq 0$, then +\begin{equation}\label{eqt10} + y\in B(x_1, D^{s_1})\, . +\end{equation} +Moreover, +\begin{equation}\label{eqt11} + \|\varphi\|_{C^\tau(B(x_1, 5D^{s_1})}\le +\frac{2^{154 a^3}}{\mu(B(x_1, D^{s_1}))\mu(B(x_2, D^{s_2}))} + \, . +\end{equation} + +\end{lemma} +\begin{proof} + +If $\varphi(y)$ is not zero, then $K_{s_1}(x_1, y)$ is not zero and thus +\eqref{supp Ks} gives \eqref{eqt10}. + +We next have for $y$ with \eqref{eq Ks size} +\begin{equation}\label{suppart} + |\varphi(y)|\le + \frac{2^{204 a^3}}{\mu(B(x_1, D^{s_1}))\mu(B(x_2, D^{s_2}))}\, . +\end{equation} +and for $y'\neq y$ +\begin{equation} + |\varphi(y)-\varphi(y')| + \end{equation} + \begin{equation} + \le + |K_{s_1}(x_1,y)-K_{s_1}(x_1,y'))|| + K_{s_2}(x_2, y)| +\end{equation} + \begin{equation}+| \overline{K_{s_1}(x_1, y')}| + |K_{s_2}(x_2, y) - K_{s_2}(x_2, y'))| +\end{equation} +\begin{equation} + \le \frac{2^{152 a^3}}{\mu(B(x_1, D^{s_1}))\mu(B(x_2, D^{s_2}))} + \left(\left(\frac{ \rho(y,y')}{D^{s_1}}\right)^{\tau}+ + \left(\frac{ \rho(y,y')}{D^{s_2}}\right)^{\tau}\right) +\end{equation} +\begin{equation}\label{holderpart} + \le \frac{2^{153 a^3}}{\mu(B(x_1, D^{s_1}))\mu(B(x_2, D^{s_2}))} + \left(\frac{ \rho(y,y')}{D^{s_1}}\right)^{\tau} +\end{equation} +Adding the estimates \eqref{suppart} and \eqref{holderpart} gives \eqref{eqt11}. +This proves the lemma. +\end{proof} +The next lemma is a geometric estimate for two tiles. +\begin{lemma}\label{lem tgeo} + Let $\fp_1, \fp_2\in \fP$ with +$\ps({\fp_1})\leq \ps({\fp_2})$. For each $x_1\in E(\fp_1)$ and +$x_2\in E(\fp_2)$ we have +\begin{equation}\label{tgeo} + 1+d_{\fp_1}(\fcc(\fp_1), \fcc(\fp_2))\le + 2^{5a}(1 + d_{B(x_1, D^{\ps(\fp_1)})}(\tQ(x_1),\tQ(x_2)))\, . +\end{equation} +\end{lemma} +\begin{proof} +Let $i\in \{1,2\}$. +By Definition \eqref{defineep} of $E$, +we have $\tQ(x_i)\in \fc(\fp_i)$ +With \eqref{eq freq comp ball} we then conclude +\begin{equation}\label{dponetwo} + d_{\fp_i}(\tQ(x_i),\fcc(\fp_i))\le 1\, . +\end{equation} +We have $\sc(\fp_1)\subset \sc(\fp_2)$ by \eqref{dyadicproperty}. +Hence with the squeezing property \eqref{eq vol sp cube} applied twice +\begin{equation} +B(\pc(\fp_1),\frac 14 D^{\ps(\fp_1)}) +\subset B(\pc(\fp_2), 4 D^{\ps(\fp_2)}) +\end{equation} +It follows +by monotonicity of the Definition \eqref{definedE} that +\begin{equation} + d_{\fp_1}(\tQ(x_2),\fcc(\fp_2))\le + d_{ B(\pc(\fp_2), 4 D^{\ps(\fp_2)})}(\tQ(x_2),\fcc(\fp_2)) + \, . +\end{equation} +Applying the doubling property \eqref{firstdb} four times and then using \eqref{dponetwo} estimates the last display by +\begin{equation}\label{tgeo0.5} + \le 2^{4a} d_{\fp_2}(\tQ(x_2),\fcc(\fp_2))\le 2^{4a} + \, . +\end{equation} +By the triangle ineqality, we obtain from \eqref{dponetwo} and +\eqref{tgeo0.5} +\begin{equation}\label{tgeo1} + 1+d_{\fp_1}(\fcc(\fp_1), \fcc(\fp_2))\le 2+2^{4a} +d_{\fp_1}(\tQ(x_1), \tQ(x_2))\, . +\end{equation} +As $x_1\in \sc(\fp_1)$ by Definition \eqref{defineep} of $E$, we have by the squeezing property \eqref{eq vol sp cube} +\begin{equation} + d(x_1,\pc(\fp_1))\le 4D^{\ps(\fp_1)} +\end{equation} +and thus by \eqref{eq vol sp cube} again and the triangle inequality +\begin{equation} + \sc(\fp_1)\subset B(x_1,8D^{\ps(\fp_1)})\, . +\end{equation} +We thus estimate the right-hand side of \eqref{tgeo1} with monotonicity of the Definition \eqref{definedE} by +\begin{equation}\label{tgeo1.5} + \le 2+2^{4a}+d_{B(x_1,8D^{\ps(\fp_1)})}(\tQ(x_1), \tQ(x_2))\, . +\end{equation} +This is further estimated by aplying the doubling property \eqref{firstdb} three times by +\begin{equation}\label{tgeo2} + \le 2+2^{4a}+2^{3a}d_{B_1(x_1, D^{s(\fp_1)})}(\tQ(x_1), \tQ(x_2))\, . +\end{equation} +Now \eqref{tgeo} follows with $a\ge 1$. +\end{proof} + + + + +\begin{lemma}\label{lem tstarsupport} + For each $\fp\in \fP$, and each $y\in X$, we have that +\begin{equation}\label{tstargnot0} + T_{\fp} g^*(y)\neq 0 +\end{equation} + implies +\begin{equation}\label{ynotfar} + y\in B(\pc(\fp),5D^{\ps(\fp)})\, . +\end{equation} +\end{lemma} +\begin{proof} +Fix $\fp$ and $y$ with \eqref{tstargnot0}. +Then there exists $x\in E(\fp)$ with +\begin{equation} + \overline{K_{\ps(\fp)}(x,y)}\overline{\tQ(x)(y)} + \tQ(x)(x)g(x) \neq 0\, . +\end{equation} +As $E(\fp)\subset \sc(\fp)$ and by the squeezing property +\eqref{eq vol sp cube}, we have +\begin{equation} + \rho(x,\pc(\fp))\le 4D^{\ps(\fp)}\, . +\end{equation} +As $K_{\ps(\fp)}(x,y)\neq 0$, we have by \eqref{supp Ks} +that +\begin{equation} +\rho(x,y)\le \frac 12 D^{\ps(\fp)}\, . +\end{equation} +Now \eqref{ynotfar} follows by the tirangle inequality. +\end{proof} + + +We now prove Lemma \ref{lem basic TT*}. We begin with \eqref{eq basic TT* est} + +We expand the left-hand side of \eqref{eq basic TT* est} as +\begin{equation}\label{tstartstar} +\left|\int \int_{E(\fp_1)} \overline{K_{\ps(\fp_1)}(x_1,y)}\overline{\tQ(x_1)(y)} + \tQ(x_1)(x_1)g(x_1)\, d\mu(x_1) \right. +\end{equation} +\begin{equation} + \times \left.\int_{E(\fp_2)} {K_{\ps(\fp_2)}(x_2,y)}{\tQ(x_2)(y)} + \overline{\tQ(x_2)(x_2)}\overline{g(x_2)}\, d\mu(x_2)\, d\mu(y)\right|\, . +\end{equation} + +By Fubini and the triangle inequality and +the fact $|\tQ(x_i)(x_i)|=1$ for $i=1,2$, we can estimate +\eqref{tstartstar} from above by +\begin{equation}\label{eqa1} +\int_{E(\fp_1)} \int_{E(\fp_2)} |\int +\overline{\tQ(x_1)(y)}{\tQ(x_2)(y)}\varphi_{x_1,x_2}(y) +\,dy|\,|g(x_1)g(x_2)|\, dx_1dx_2\,. +\end{equation} +We estimate for fixed $x_1\in E(\fp_1)$ and +$x_2\in E(\fp_2)$ the inner integral of \eqref{eqa1} with +Proposition \ref{lem vdc regularity}. The function +$\varphi:=\varphi_{x_1,x_2}$ satisfies the assumptions of +Proposition \ref{lem vdc regularity} by Lemma \ref{lem ksquare}. +We obtain with +\begin{equation} + B':= B(x_1, D^{\ps(\fp_1)})\, , +\end{equation} +\begin{equation} + |\int +\overline{\tQ(x_1)(y)}{\tQ(x_2)(y)}\varphi_{x_1,x_2}(y) +\,dy| +\end{equation} +\begin{equation} + \le 2^{4a} \mu(B') \|{\varphi}\|_{C^\tau(B')} + (1 + d_{B'}(\tQ(x_1),\tQ(x_2)))^{-\tau^2/(2+a)} +\end{equation} +\begin{equation} + \le \frac{2^{154a^3}} + {\mu(B(x_2, D^{\ps(\fp_2)}))} + (1 + d_{B'}(\tQ(x_1),\tQ(x_2)))^{-\tau^2/(2+a)} +\end{equation} +Using Lemma \ref{lem tgeo} and $a\ge 1$ and $\tau \le 1$ estimates the last display by +\begin{equation}\label{eqa2} + \le \frac{2^{159a^3}} + {\mu(B(x_2, D^{\ps(\fp_2)}))} + (1+d_{\fp_1}(\fcc(\fp_1), \fcc(\fp_2)))^{-\tau^2/(2+a)} +\end{equation} +As $x_2\in \sc(\fp_2)$ by Definition \eqref{defineep} of $E$, we have by \eqref{eq vol sp cube} +\begin{equation} + d(x_2,\pc(\fp_2))\le 4D^{\ps(\fp_2)} +\end{equation} +and thus by \eqref{eq vol sp cube} again and the triangle inequality +\begin{equation} + \sc(\fp_2)\subset B(x_2,8D^{\ps(\fp_2)})\, . +\end{equation} +Using three iterations of the doubling property \eqref{doublingx} give +\begin{equation} + \mu(\sc(\fp_2))\le 2^{3a}\mu(B(x_2,D^{\ps(\fp_2)}))\, . +\end{equation} +With $a\ge 1$ and \eqref{eqa2} we conclude \eqref{eq basic TT* est} and thus complete the proof of the lemma. + + +Now assume the left-hand side of \eqref{eq basic TT* est} is not zero. +There is a $y\in X$ with +\begin{equation} + T^*_{\fp}g(y)\overline{T^*_{\fp'}g(y)}\neq 0 +\end{equation} +By the triangle inequality and Lemma \ref{lem tstarsupport}, we conclude +\begin{equation} + \rho(\pc(\fp),\pc(\fp'))\le \rho(\pc(\fp),y) +\rho(\pc(\fp'),y) + \le 5D^{\ps(\fp)}+5D^{\ps(\fp')}\le 10 D^{\ps(\fp)}\, . +\end{equation} +By the squeezing property \eqref{eq vol sp cube} and the triangle inequality, +we conclude +\begin{equation} + \sc(\fp') \subset B(\pc(\fp), 15D^{\ps(\fp)})\, . +\end{equation} + This completes the proof of Lemma \ref{lem basic TT*}. + + + + + +\section{Proof of Lemma \ref{lem antichain 1}, the geometric estimate} +\label{subsec geolem} + + +\begin{lemma}\label{lem a geo} +Let $\mfa\in \Mf$ and $N\ge0$ be an integer. +Let $\fp, \fp'\in \fP$ with +\begin{equation}\label{eqassumedismfa} + d_{\fp}(\fcc(\fp), \mfa))\le 2^N\, +\end{equation} +\begin{equation}\label{eqassumedismfap} + d_{\fp'}(\fcc(\fp'), \mfa))\le 2^N\, . +\end{equation} +Assume $\sc(\fp)\subset \sc(\fp')$ and $\ps(\fp)<\ps(\fp')$. +Then +\begin{equation}\label{lp'lp''}2^{4a+N+2}\fp\lesssim 2^{4a+N+2} \fp'\, . +\end{equation} +\end{lemma} + +\begin{proof} + By the squeezing property \eqref{eq freq comp ball} + and by assumption, we have +\begin{equation}\label{ageo0} + B(\pc(\fp), \frac 14 D^{\ps(\fp)})\subset \sc(\fp)\subset \sc(\fp') + \subset B(\pc(\fp'), 4D^{\ps(\fp')})\, . +\end{equation} + Applying the doubling property \eqref{firstdb} four times, and the monotonicity in + the set in Definition \eqref{definedE} gives with + \eqref{eqassumedismfap} +\begin{equation} + d_{\fp}(\fcc(\fp'),\mfa) + \le 2^{4a} d_{\fp'}(\fcc(\fp'),\mfa)' + \le 2^{4a+N} \, . +\end{equation} +Together with \eqref{eqassumedismfa} and the triangle inequality, we obtain +\begin{equation}\label{eqdistqpqp} + d_{\fp'}(\fcc(\fp'),\fcc(\fp'')\le 2^{4a+N+1} \, . +\end{equation} +Now assume +\begin{equation} + \mfa'\in B_{\fp'}(\fcc(\fp'),2^{4a+N+2}). +\end{equation} +By the doubling property \eqref{firstdb}, applied five times, we have +\begin{equation}\label{ageo1} d_{B(\pc(\fp'),8D^{\ps(\fp')})}(\fcc(\fp'),\mfa') < 2^{9a+N+2}\, . +\end{equation} +We have by \eqref{ageo0} +\begin{equation} + \pc(\fp)\in +B(\pc(\fp'),4D^{\ps(\fp')})\, . +\end{equation} +Hence by the triangle inequality +\begin{equation} + B(\pc(\fp), 4D^{\ps(\fp')}) + \subset +B(\pc(\fp'),8D^{\ps(\fp')})\, . +\end{equation} +Together with \eqref{ageo1} and monotonicity of the Definition \eqref{definedE} +of $d_E$, +\begin{equation} + d_{B(\pc(\fp),4D^{\ps(\fp')})}(\fcc(\fp'),\mfa') < 2^{9a+N+2}\, . +\end{equation} +Using the doubling property \eqref{seconddb} $5a+2$ times gives +\begin{equation} + d_{B(\pc(\fp),2^{2-5a^2-a}D^{\ps(\fp')})}(\fcc(\fp'),\mfa') < 2^{4a+N}\, . +\end{equation} +Using $\ps(\fp')<\ps(\fp'')$ and $D=2^{100a^2}$ and $a\ge 4$ gives +\begin{equation} + d_{\fp}(\fcc(\fp'),\mfa') < 2^{4a+N}\, . +\end{equation} +With the triangle inequality and \eqref{eqdistqpqp}, +\begin{equation} + d_{\fp}(\fcc(\fp),\mfa') < 2^{4a+N+2}\, . +\end{equation} +This shows +\begin{equation} +B_{\fp'}(\fcc(\fp'),2^{4a+N+2})\subset B_{\fp}(\fcc(\fp),2^{4a+N+2})\, . +\end{equation} +This implies \eqref{lp'lp''} and completes the proof of the lemma. + +\end{proof} + +For $\mfa \in \Mf$ and $N\ge 0$ define +\begin{equation}\label{eqantidefap} + \mathfrak{A}_{\mfa,N}:=\{\fp\in\mathfrak{A}: 2^{N}\le 1+d_{\fp}(\fcc(\fp), \mfa))\le 2^{N+1}\} \, . +\end{equation} + + +\begin{lemma}\label{lem samel} +Let $\mfa \in \Mf$ and $N\ge 0$ and +$L\in \mathcal{D}$. Then +\begin{equation}\label{eqanti-1} + \sum_{\fp\in\mathfrak{A}_{\mfa,N}:\sc(\fp)=L}\mu(E(\fp)\cap G)\le 2^{a(N+5)}\dens_1(\mathfrak{A})\mu(L)\, . +\end{equation} +\end{lemma} +\begin{proof} +Let $\mfa,N,L$ be given and set +\begin{equation} +\mathfrak{A}':=\{\fp\in\mathfrak{A}_{\mfa,N}:\sc(\fp)=L\}\, . +\end{equation} + + + +Let +$\fp\in\mathfrak{A}'$. +We have +by Definition \eqref{definedens1} +using $\lambda=2$ and the squeezing property \eqref{eq freq comp ball} +\begin{equation}\label{eqanti-3} +\mu(E(\fp)\cap G)\le \mu(E_2(2, \fp))\le 2^{a}\dens_1(\mathfrak{A})\mu(L)\, . +\end{equation} +By the covering property \eqref{thirddb}, applied $N+4$ times, there is a collection $\Mf'$ of at most $2^{a(N+4)}$ +elements such that +\begin{equation}\label{eqanti-4} + B_{\fp}(\mfa, 2^{N+1})\subset \bigcup_{\mfa'\in MF'} + B_{\fp}(\mfa', 0.2)\, . +\end{equation} +As each $\fcc(\fp)$ with $\fp\in \mathfrak{A}_{\mfa,N}$ +is contained in the left-hand-side +of \eqref{eqanti-4} +by definition, it is in at least one $B_{\fp}(\mfa', 0.2)$ +with $\mfa'\in \Mf'$. + + +For two different $\fp,\fp'\in \mathfrak{A}'$, we have by +\eqref{eq dis freq cover} that +$\fc(\fp)$ and $\fc(\fp')$ are disjoint and thus by the squeezing property \eqref{eq freq comp ball} we have for every $\mfa'\in \Mf'$ +\begin{equation} + \mfa'\not\in B_{\fp}(\fcc(\fp), 0.2)\cap +B_{\fp}(\fcc(\fp'), 0.2)\, . +\end{equation} +Hence at most one of $\fcc(\fp)$ +and $\fcc(\fp)$ is in +$B_{\fp}(\mfa', 0.2)$. +It follows that there are at most $2^{a(N+4)}$ elements in +$\mathfrak{A}'$. Adding \eqref{eqanti-3} over $\mathfrak{A}'$ proves +\eqref{eqanti-1}. + + +\end{proof} + + +\begin{lemma}\label{lem antichain-.5} +Let $\mfa\in\Mf$ and be +an integer. Let $\fp_\mfa$ be a tile with $\mfa\in \fc(\fp_\mfa)$. +Then we have +\begin{equation}\label{eqanti-0.5} + \sum_{\fp\in\mathfrak{A}_{\mfa,N}: \ps(\fp_\mfa)<\ps(\fp)}\mu(E(\fp)\cap G \cap \sc(\fp_\mfa)) + \le \mu (E_2(2^{4a+N+3},\fp_\mfa)) + \, . +\end{equation} + + + +\end{lemma} + +\begin{proof} + + +Let $\fp$ be any tile in $\mathfrak{A}_{\mfa,N}$ with $\ps(\fp_\mfa)<\ps(\fp)$. By definition of +$E$, the tile contributes zero to the sum on the left-hand side of \eqref{eqanti-0.5} unless + $\sc(\fp)\cap \sc(\fp_{\mfa}) \neq \emptyset$, which we may assume. With $\ps(\fp_\mfa)<\ps(\fp)$ +and the dyadic property +\eqref{dyadicproperty} we conclude $\sc(\fp_{\mfa})\subset \sc(\fp)$. +By the squeezing property +\eqref{eq freq comp ball}, +we conclude from +$\mfa\in \fc(\fp_\mfa)$ +that +\begin{equation} + \mfa\in B(\fcc(\fp_\mfa), 1)\, . +\end{equation} +We conclude from $\fp \in \mathfrak{A}_{\mfa,N}$ that +\begin{equation} + \mfa \in B(\fcc(\fp), 2^{N+1})\, . +\end{equation} +With Lemma \ref{lem a geo}, we conclude + \begin{equation} + 2^{4a+N+3}\fp_\mfa \lesssim 2^{4a+N+3}\fp \, . + \end{equation} +By the squeezing property \eqref{eq freq comp ball} +and $a\ge 1$ and $N\ge 0$, we conclude +\begin{equation} + \fcc(\fp)\subset B(2^{4a+N+1}, \fcc(\fp_\mfa)\, . +\end{equation} +By Definition \eqref{definee2} of $E_2$, + we conclude +\begin{equation} +E(\fp)\cap G \subset E_2(2^{4a+N+3},\fp_\mfa)\, . +\end{equation} +Using disjointness of the various $E(\fp)$ with $\fp\in \mathfrak{A}$ by Lemma \ref{lem antichain -1}, we obtain \eqref{eqanti-0.5}. +This proves the lemma. +\end{proof} +\begin{lemma} +\label{lem antichain 0} +Let $\mfa\in\Mf$ and let $N\ge 0$ be +an integer. Then we have +\begin{equation}\label{eqanti00} + \sum_{\fp\in\mathfrak{A}_{\mfa,N}}\mu(E(\fp)\cap G) + \le + 2^{101a^3+Na}\dens_1(\mathfrak{A})\mu\left(\cup_{\fp\in\mathfrak{A}}I_{\fp}\right)\, . +\end{equation} +\end{lemma} + + + +\begin{proof} + Fix $\mfa$ and $N$ and let +$\mathfrak{A}'$ for the set of $\fp\in=\mathfrak{A}_{\mfa,N}$ such that $\sc(\fp)\cap G$ is not empty. + + + + Let $\mathcal{L}$ be the collection dyadic cubes $I\in\mathcal{D}$ such that $I\subset \sc(\fp)$ for some $\fp\in\mathfrak{A}'$ and if $\sc(\fp)\subset I$ for some $\fp\in\mathfrak{A}'$, then $\ps(\fp)=-S$. By \eqref{coverdyadic}, for each $\fp \in \mathfrak{A}'$ + and each $x\in \sc(\fp)\cap G$, there is a $I\in \mathcal{D}$ with $s(I)=-S$ and $x\in I$. By \eqref{dyadicproperty}, + we have $I\subset \sc(\fp)$. Hence + \begin{equation} + \sc(\fp)\subset \bigcup\{I\in \mathcal{D}: s(I)=-S, I\subset \sc(\fp)\}\subset \bigcup \mathcal{L}\, . + \end{equation} +As each $I\in \mathcal{L}$ satisfies $I\subset \sc(\fp)$ for some $\fp$ in $\mathfrak{A'}$, we conclude + \begin{equation} +\bigcup\mathcal{L}=\bigcup_{\fp \in \mathfrak{A}'}\sc(\fp)\, . + \end{equation} +Let $\mathcal{L}^*$ be the set of maximal elements on $\mathcal{L}$ with respect to set inclusion. +By \eqref{dyadicproperty}, the elements in $\mathcal{L}^*$ are pairwise disjoint and we have + \begin{equation}\label{eqdecAprime} +\bigcup\mathcal{L}^*=\bigcup_{\fp \in \mathfrak{A}'}\sc(\fp)\, . + \end{equation} +Using the partition \eqref{eqdecAprime} into elements of $\mathcal{L}$ in \eqref{eqanti0}, it suffices to show for each $L\in \mathcal{L}^*$ +\begin{equation}\label{eqanti0} + \sum_{\fp\in\mathfrak{A}'}\mu(E(\fp)\cap G \cap L) + \le +2^{101a^3+aN} +\dens_1(\mathfrak{A})\mu(L)\, , +\end{equation} +Fix $L\in \mathcal{L}^*$. +By definition of $L$, there exists an element $\fp'\in \mathfrak{A}'$ such that $L\subset \sc(\fp')$. Pick such an element $\fp +'$ +in $\mathfrak{A}$ with minimal $\ps(\fp')$. As $\sc(\fp')\not \subset L$ by definition of $L$, we have +with \eqref{dyadicproperty} that $s(L)< \ps(\fp')$. In particular $s(L)4$ gives \eqref{eqanti0}. +This completes the proof of the lemma. + + + + + + + +\end{proof} + + + +We turn to the proof of Lemma \ref{lem antichain 1}. + + + + +\begin{proof} + + +Using that $\mathfrak{A}$ is the union of the +$\mathfrak{A_{\mfa,N}}$ with $N\ge 0$, +we estimate the left-hand side of +with the triangle inequality by +\begin{equation}\label{eqanti23} +\le \sum_{N\ge 0} \left\|\sum_{\fp\in \mathfrak{A}_{\mfa,N}} 2^{-N\tau^2/(2+a)}1_{E(\fp)} 1_G\right\|_{p} +\end{equation} +We consider each individual term in this sum and estimate it's $p$-th power. + Using that for each $x\in X$ by Lemma \ref{lem antichain 0} there is at most one $\fp\in \mathfrak{A}$ with $x\in E(\fp)$, + we have + \begin{equation} + \left\|\sum_{\fp\in \mathfrak{A}_{\mfa,N}} 2^{-N\tau^2/(2+a)}1_{E(\fp)} 1_G\right\|_{p}^p + \end{equation} +\begin{equation} + =\int_G(\sum_{\fp\in\mathfrak{A}}2^{-N\tau^2/(2+a)}\mathbf{1}_{E(\fp)}(x))^p\, d\mu(x) +\end{equation} +\begin{equation} + = \int _G\sum_{\fp\in\mathfrak{A}}2^{-(p-1)N\tau^2/(2+a)}\mathbf{1}_{E(\fp)}(x)\, d\mu(x) +\end{equation} +\begin{equation} + = 2^{-(p-1)N\tau^2/(2+a)} \sum_{\fp\in\mathfrak{A}_{\mfa,N}}\mu(E(\fp)\cap G) +\end{equation} + +Using Lemma \ref{lem antichain 0}, we estimate the last display by +\begin{equation}\label{eqanti21} +\le 2^{-(p-1)N\tau^2/(2+a)+101a^3+Na}\dens_1(\mathfrak{A})\mu\left(\cup_{\fp\in\mathfrak{A}}\sc(\fp)\right) +\end{equation} +Using that with $a\ge 4$ and $0<\tau\le 1$ we have +\begin{equation} +(p-1)N\tau^2/(2+a)\ge +(p-1)N\tau^2/(2a)\ge (2a -1)N\ge Na+N\, . +\end{equation} +Hence we have for \eqref{eqanti21} the upper bound +\begin{equation}\label{eqanti22} +\le 2^{101a^3-N}\dens_1(\mathfrak{A})\mu\left(\cup_{\fp\in\mathfrak{A}}\sc(\fp)\right)\, . +\end{equation} +Taking th $p$-th root and summing over $N\ge 0$ gives for \eqref{eqanti23} the upper bound + +\begin{equation} +\le \left(\sum_{N\ge 0} 2^{-N/p}\right)2^{101a^3/p}\dens_1(\mathfrak{A})\mu\left(\cup_{\fp\in\mathfrak{A}}\sc(\fp)\right) +\end{equation} +\begin{equation} +\le \left(1-2^{-1/p}\right)^{-1} +2^{101a^3/p} +\dens_1(\mathfrak{A})^{\frac 1p}\mu\left(\cup_{\fp\in\mathfrak{A}}\sc(\fp)\right)^{\frac 1p}\, . +\end{equation} +This proves the lemma. +\end{proof} + + + + + + + +\chapter{Proof of Proposition \ref{forestprop}} + +\label{treesection} + +\section{Tree Estimate} +%%%%%% Previous discussion about non-tangential Cotlar maximal inequality +% Given a measurable function $\sigma$ mapping $X$ to the set of finite convex subsets of $\mathbb{Z}^d$, we define an associated truncated singular maximal operator +% \begin{equation} +% \label{def Tsigma} +% T_\sigma f(x):=\sum_{s\in\sigma(x)}\int K_s(x,y) f(y)\,dy. +% \end{equation} +% From the proof of Cotlar's inequality, it can be seen that the non-tangentially maximally truncated operator +% \begin{equation} +% \label{def non-tang max op} +% T_{\mathcal{N}}f(x):=\sup_{\rho(x,x') s(L)$ for all $\fp \in \fT(\fu)$ with $L \cap \sc(\fp) \ne \emptyset$. In particular, we have that $\underline{\sigma}(\fu, x) > s(L)$. By \eqref{eq vol sp cube}, we have $\rho(x,x') \le 8D^{s(L)} \le 8D^{\underline{\sigma}(\fu, x) - 1}$, and by Lemma \ref{lem sigma convex} the set $\sigma(\fu, x)$ is an interval. Hence, \lars{write correct name of nontangential operator} + $$ + \eqref{eq term B} \le T_{\mathcal{N}, } P_{\mathcal{J}(\fT(\fu))} f(x')\,. + $$ + + Next, we control \eqref{eq term A}. If $K_s(x,y)\neq 0$, then by \eqref{eq Ks supp t} $\rho(x,y)\leq 1/2 D^s$. By $1$-Lipschitz continuity of the function $t \mapsto \exp(it) = e(t)$, it follows that + \begin{multline*} + |e(-Q(\fu)(y)+Q(x)(y)+Q(\fu)(x)-Q(x)(x))-1|\\ + \leq d_{B(x, 1/2 D^{s})}(Q(\fu), Q(x))\,. + \end{multline*} + Let $\fp_s \in \fT(\fu)$ be a tile with $s(\fp_s) = s$ and $x \in E(\fp_s)$, and let $\fp'$ be a tile with $s(\fp') = \overline{\sigma}(\fu, x)$ and $x \in E(\fp')$. + Using \eqref{firstdb}, \eqref{eq vol sp cube} and Lemma \ref{lem cube monotone}, we obtain + $$ + \le 2^a d_{\fp_s}(Q(\fu), Q(x)) \le 2^{a} 2^{s - \overline{\sigma}(\fu, x)} d_{\fp'}(Q(\fu), Q(x))\,. + $$ + Since $Q(\fu) \in B_{\fp'}(Q(\fp'), 4)$ by \eqref{forest1} and $Q(x) \in \Omega(\fp') \subset B_{\fp'}(Q(\fp'), 1)$ by \eqref{eq freq comp ball}, we can estimate this by + $$ + \le 5 \cdot 2^{a} 2^{s - \overline{\sigma}(\fu, x)} \,. + $$ + Using \eqref{eq Ks size}, it follows that + $$ + \eqref{eq term A} \le 5\cdot 2^{103a^3} \sum_{s\in\sigma(x)}2^{s - \overline{\sigma}(\fu, x)} \frac{1}{\mu(B(x,D^s))}\int_{B(x,0.5D^{s})}|f(y)|\,\mathrm{d}\mu(y)\,. + $$ + Since the collection $\mathcal{J}$ is a partition of $X$, we can estimate this by + $$ + 5\cdot 2^{103a^3} \sum_{s\in\sigma(x)}2^{s - \overline{\sigma}(\fu, x)} \frac{1}{\mu(B(x,D^s))}\sum_{\substack{J \in \mathcal{J}(\fT(\fu))\\J \cap B(x, 0.5D^s) \ne \emptyset} }\int_{J}|f(y)|\,\mathrm{d}\mu(y)\,. + $$ + This expression does not change if we replace $|f|$ by $P_{\mathcal{J}(\fT(\fu))}|f|$. Further, if $J \in \mathcal{J}(\fT(\fu))$ with $B(x, 0.5 D^s) \cap J \ne \emptyset$, then $B(c(\fp_s), 4.5D^s) \cap J \ne \emptyset$ by the triangle inequality. If $s(J) \ge s - 1$, then it follows from the triangle inequality, \eqref{eq vol sp cube} and \eqref{defineD} that $\sc(\fp) \subset B(c(J), 100 D^{s(J)+1})$, contradicting $J \in \mathcal{J}(\mathfrak{T}(\fu))$. Thus $s(J) < s - 1$. By the triangle inequality and \eqref{eq vol sp cube}, we conclude that $J \subset B(x, D^s)$. Thus we can continue our chain of estimates with + $$ + 5\cdot 2^{103a^3} \sum_{s\in\sigma(x)}2^{s - \overline{\sigma}(\fu, x)} \frac{1}{\mu(B(x,D^s))}\int_{B(x,D^s)}|P_{\mathcal{J}(\fT(\fu))}f(y)|\,\mathrm{d}\mu(y)\,. + $$ + We have $B(x, D^s) \subset B(\sc(\fp_s), 8D^s)$ and $B(\sc(\fp_s), 8D^s)) \subset B(x, 16D^s)$, both by \eqref{eq vol sp cube} since $x \in \sc(\fp)$. Combining this with the doubling property \eqref{doublingx}, we obtain + $$ + \mu(B(c(\fp_s), 8D^s)) \le 2^{4a} \mu(B(x, D^s))\,. + $$ + Hence \eqref{eq term A} is bounded by + $$ + 5\cdot 2^{107a^3} \sum_{s\in\sigma(x)}2^{s - \overline{\sigma}(\fu, x)} \frac{1}{\mu(B(c(\fp_s),8D^s))}\int_{B(c(\fp_s),8D^s)}|P_{\mathcal{J}(\fT(\fu))}f(y)|\,\mathrm{d}\mu(y)\,. + $$ + Since $\rho(x,x') \le 8 D^{s - 1}$, as shown above, we have $x' \in B(x, D^s) \subset B(\sc(\fp_s), 8D^s)$. Thus + $$ + \le 5\cdot 2^{107a^3} \sum_{s\in\sigma(x)}2^{s - \overline{\sigma}(\fu, x)} M_{\mathcal{B}, 1}|P_{\mathcal{J}(\fT(\fu))}f|(x') + $$ + $$ + \le 10 \cdot 2^{107a^3} M_{\mathcal{B}, 1}|P_{\mathcal{J}(\fT(\fu))}f|(x')\,. + $$ + This completes the estimate for term \eqref{eq term A}. + + + Finally, we estimate \eqref{eq term C}. We have for $J \in \mathcal{J}(\fT(\fu))$: + $$ + \int_J K_{s}(x,y)(1 - P_{\mathcal{J}(\fT(\fu))})f(y) \, \mathrm{d}\mu(y) + $$ + \begin{equation} + \label{eq canc comp} + = \int_J \frac{1}{\mu(J)} \int_J K_s(x,y) - K_s(x,z) \, \mathrm{d}\mu(z) \,f(y) \, \mathrm{d}\mu(y)\,. + \end{equation} + By \eqref{eq Ks smooth} and \eqref{eq vol sp cube}, we have for $y, z \in J$ + $$ + |K_s(x,y) - K_s(x,z)| \le \frac{2^{150a^3}}{\mu(B(x, D^s))} \left(\frac{8 D^{s(J)}}{D^s}\right)^\tau\,. + $$ + Using this, \eqref{eq canc comp}, and the fact, shown in the estimate for term \eqref{eq term A}, that $J \subset B(x, D^s)$ if it intersects the support of $K_s(x,y)$, we estimate \eqref{eq term C} by + $$ + 2^{150a^3 + 3\tau}\sum_{\fp\in \mathfrak{T}}\frac{\mathbf{1}_{E(\fp)}}{\mu(B(x,D^{s(\fp)}))}(x)\sum_{\substack{J\in \mathcal{J}\\J\subset B(x, D^{s(\fp)})}} D^{\tau(s(J) - s(\fp))} \int_J |f|. + $$ + $$ + = 2^{150a^3 + 3\tau}\sum_{I \in \mathcal{D}} \sum_{\substack{\fp\in \mathfrak{T}\\ \sc(\fp) = I}}\frac{\mathbf{1}_{E(\fp)}(x)}{\mu(B(x, D^{s(I)}))}\sum_{\substack{J\in \mathcal{J}\\J\subset B(x, D^{s(\fp)})}} D^{\tau(s(J) - s(\fp))} \int_J |f|. + $$ + By \eqref{eq dis freq cover} and \eqref{definedE}, the sets $E(\fp)$ for tiles $\fp$ with $\sc(\fp) = I$ are pairwise disjoint. + If $x \in E(\fp)$ then in particular $x \in \sc(\fp)$, so by \eqref{eq vol sp cube} $B(c(I),8D^{s(I)}) \subset B(x, 16D^{s(I)})$. By the doubling property \eqref{doublingx} + $$ + \mu(B(c(I), D^{s(I)})) \le 2^{4a} \mu(B(x, D^{s(I)}))\,. + $$ + Hence we can complete our estimate with + $$ + \le 2^{157a^3}\sum_{I \in \mathcal{D}} \frac{\mathbf{1}_{I}(x)}{\mu(B(c(I), 8D^{s(I)}))}\sum_{\substack{J\in \mathcal{J}\\J\subset B(x, D^{s(\fp)})}} D^{\tau(s(J) - s(\fp))} \int_J |f|\,. + $$ + Finally, it follows from the definition of $\mathcal{L}(\fT(\fu))$ that $x \in \sc(\fp)$ if and only if $x' \in \sc(\fp)$, thus this equals + $$ + 2^{157a^3} S P_{\mathcal{J}(\fT(\fu))}|f|(x')\,. + $$ + This completes the proof. +\end{proof} + +\begin{lemma} + \label{lem aux overlap} + For every cube $I \in \mathcal{D}$, there exist at most $2^{4a}$ cubes $J \in \mathcal{D}$ with $s(J) = s(I)$ and $B(c(I), D^{s(I)}) \cap B(c(J), D^{s(J)}) \ne \emptyset$. +\end{lemma} + +\begin{proof} + Suppose that $B(c(I), D^{s(I)}) \cap B(c(J), D^{s(J)}) \ne \emptyset$ and $s(I) = s(J)$. Then $B(c(I), 2D^{s(I)}) \subset B(c(J), 4D^{s(J)})$. Hence by the doubling property \eqref{doublingx} + $$ + 2^{4a}\mu(B(c(J), \frac{1}{4}D^{s(J)})) \ge \mu(B(c(I), 2D^{s(I)}))\,, + $$ + and by the triangle inequality, the ball $B(c(J), \frac{1}{4}D^{s(J)})$ is contained in $B(c(I), 2D^{s(I)})$. If $\mathcal{C}$ is any finite collection of cubes $J \in \mathcal{D}$ satisfying $s(J) = s(I)$ and $B(c(I), D^{s(I)}) = B(c(J), D^{s(J)})$, then it follows from \eqref{eq vol sp cube} and pairwise disjointness of cubes of the same scale \eqref{dyadicproperty} that the balls $B(c(J), \frac{1}{4} D^{s(J)})$ are pairwise disjoint. Hence + $$ + \mu(B(c(I), 2D^{s(I)})) \ge \sum_{J \in \mathcal{C}} \mu(B(c(J), \frac{1}{4}D^{s(J)})) \ge |\mathcal{C}| 2^{-4a} \mu(B(c(I), 2D^{s(I)}))\,. + $$ + \lars{Maybe prove this somewhere} + It follows from the fact that $\mu$ is doubling and nonzero that $\mu(B(c(I), 2D^{s(I)})) > 0$. The lemma follows. +\end{proof} + +\begin{lemma} + For all $\fu \in \fU$ and all $f \in L^2(X)$, we have + \begin{equation} + \label{eq S bound} + \|S_{\fu}f\|_2 \le ... \|f\|_2\,. + \end{equation} +\end{lemma} + +\begin{proof} + We have for every $g \in L^2(X)$ + \begin{align*} + &\quad \big|\int g Sf\big|\\ + &= \sum_{I\in\mathcal{D}} \frac{1}{\mu(B(c(I), 8D^{s(I)}))} \int_I g(y) \, \mathrm{d}\mu(y) \sum_{J\in \mathcal{J}:J\subseteq B(c(I), D^{s(I)})} D^{(s(J)-s(I))\tau}\int_J |f(y)| \,\mathrm{d}\mu(y)\\ + &\le \sum_{I\in\mathcal{D}} \frac{1}{\mu(B(c(I), 8D^{s(I)}))} \int_{B(c(I), 8D^{s(I)})} |g(y)| \, \mathrm{d}\mu(y) \sum_{J\in \mathcal{J}:J\subseteq B(c(I), D^{s(I)})} D^{(s(J)-s(I))\tau}\int_J |f(y)| \,\mathrm{d}\mu(y)\,. + \end{align*} + Changing the order of summation and using $J \subset B(c(I), D^{s(I)})$ to bound the first average integral by $M_{\mathcal{B},1}|g|(y)$ for any $y \in J$, we obtain + \begin{align*} + \le \sum_{J\in\mathcal{J}}\int_J|f(y)| M_{\mathcal{B},1}g(y) \, \mathrm{d}\mu(y) \sum_{I \in \mathcal{D} \ : \ J\subset B(c(I), D^{s(I)})} D^{(s(J)-s(I))\tau}. + \end{align*} + By \eqref{eq vol sp cube} and \eqref{defineD} the condition $J \subset B(c(I), D^{s(I)})$ implies $s(I) \ge s(J)$. By Lemma \ref{lem aux overlap}, there are at most $2^{4a}$ cubes $I$ at each scale with $J \subset B(c(I), D^{s(I)})$. Summing the geometric series using that $D \ge e$ and hence $(1 - D^\tau)^{-1} \le \tau^{-1}$, we obtain + $$ + \le \frac{2^{4a}}{\tau} \sum_{J\in\mathcal{J}}\int_J|f(y)| M_{\mathcal{B},1}g(y) \, \mathrm{d}\mu(y)\,. + $$ + The collection $\mathcal{J}$ is a partition of $X$, so this equals + $$ + \frac{2^{4a}}{\tau} \int_X|f(y)| M_{\mathcal{B},1}g(y) \, \mathrm{d}\mu(y)\,. + $$ + Using Cauchy-Schwarz and Proposition \ref{prop hlm} we conclude + $$ + \left|\int g Sf \, \mathrm{d}\mu \right| \le 2\frac{2^{6a}}{\tau} \|g\|_2\|f\|_2\,. + $$ + The lemma now follows from duality. \lars{Is this duality lemma in Lean? Also, this proof is not really finitary, because the integrals are not. Is it necessary in Lean to prove in each step that all integrals converge?} +\end{proof} + +\begin{lemma}[Tree estimate] + \label{TreeEstimate} + Let $\fu \in \fU$. + Then we have for all $f, g$ + \begin{equation} + \label{eq tree est} + \int |\sum_{\fp \in \fT(\fu)} gT_{\fp}f \, \mathrm{d}\mu |\le ... \|P_{\mathcal{J}(\fT(\fu))}|f|\|_{2}\|P_{\mathcal{L}(\fT(\fu))}|f|\|_{2}. + \end{equation} +\end{lemma} + +\begin{proof} + Combine the last two lemmas. +\end{proof} + +\printbibliography diff --git a/blueprint/src/preamble/common.tex b/blueprint/src/preamble/common.tex index c0ca82b6..a9aa55a4 100644 --- a/blueprint/src/preamble/common.tex +++ b/blueprint/src/preamble/common.tex @@ -1,34 +1,69 @@ +% !TeX root = print.tex % Put any macro and import needed for the project here. % This will be used by both the web and print versions of the blueprint. % This file is not meant to be built. Build src/web.tex or src/print.text instead. -% Letters -\newcommand{\C}{\mathbb{C}} -\newcommand{\bbc}{\mathbb{C}} -\newcommand{\E}{\mathbb{E}} -\newcommand*{\bbe}{\mathbb{E}} -\newcommand{\F}{\mathbb{F}} -\newcommand{\bbf}{\mathbb{F}} -\newcommand{\bbH}{\mathbb{H}} -\newcommand{\bbP}{\mathbb{P}} -\newcommand{\bbI}{\mathbb{I}} -\newcommand{\bbn}{\mathbb{N}} -\newcommand{\bbq}{\mathbb{Q}} -\newcommand{\bbr}{\mathbb{R}} -\newcommand{\bbt}{\mathbb{T}} -\newcommand{\bbz}{\mathbb{Z}} +\usepackage{mathtools} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage{amsmath} +\usepackage{graphicx} +\usepackage{xcolor} +\usepackage{enumitem} +\usepackage[hidelinks, colorlinks=false]{hyperref} +\usepackage{showlabels} + +\newcommand{\rs}[1]{{\color{blue} RS: #1.}} +\newcommand{\lars}[1]{{\color{red} LB: #1.}} +\newcommand{\asgar}[1]{{\color{TealBlue} AJ: #1}} +\newcommand{\ct}[1]{{\color{purple} CT: #1}} -\newcommand{\lo}[1]{\mathcal{L}{#1}} -% Paired delimiters -\newcommand{\abs}[1]{\left\lvert #1\right\rvert} -\newcommand{\Abs}[1]{\lvert #1 \rvert} -\newcommand{\brac}[1]{\left( #1\right)} -\newcommand{\norm}[1]{\lVert #1\rVert} -\newcommand{\inn}[1]{\left\langle #1 \right\rangle} +\theoremstyle{plain} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{prop}[theorem]{Proposition} +\newtheorem{cor}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{example}[theorem]{Example} +\newtheorem{examples}[theorem]{Examples} -% Operators -\DeclareMathOperator{\dist}{dist} +\numberwithin{theorem}{section} +\numberwithin{equation}{section} + +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\N}{\mathbb{N}} +\DeclareMathOperator{\ch}{\operatorname{ch}} +\DeclareMathOperator{\dens}{\operatorname{dens}} +\DeclareMathOperator{\supp}{\operatorname{supp}} +\DeclareMathOperator{\tp}{\operatorname{top}} +\DeclareMathOperator{\im}{\operatorname{im}} +\DeclareMathOperator{\Lip}{\operatorname{Lip}} +\DeclareMathOperator{\bd}{\operatorname{bd}} +\DeclareMathOperator*{\esssup}{\operatorname{ess\,sup}} -\newcommand{\ind}[1]{1_{#1}} -\providecommand{\tup}[1]{{\vec{#1}}} +\def \fp {\mathfrak p} +\def \fP {\mathfrak P} +\def \fu {\mathfrak u} +\def \fU {\mathfrak U} +\def \fv {\mathfrak v} +\def \fq {\mathfrak q} +\def \fQ {\mathfrak Q} +\def\fT{\mathfrak T} +\def\fL{\mathfrak L} +\def\fC{\mathfrak C} +\def\pc{\mathrm{c}} +\def\ps{\mathrm{s}} +\def \AD{{\bf s}} +\def \fc{\Omega} +\def\borel{\mathcal{E}} +\def\borelb{\mathcal{G}} +\def\sc{\mathcal{I}} +\def \tQ{{Q}} +\def\mfa{\vartheta} +\def\mfb{\theta} +\def\Mf{\Theta} +\def\fcc{\mathcal{Q}} diff --git a/blueprint/src/preamble/print.tex b/blueprint/src/preamble/print.tex index 013b983e..a677b7bb 100644 --- a/blueprint/src/preamble/print.tex +++ b/blueprint/src/preamble/print.tex @@ -1,14 +1,6 @@ % Those macros are used for the print version of the blueprint. % This file is not meant to be built. Build src/web.tex or src/print.text instead. -\declaretheorem[numberwithin=chapter]{theorem} -\declaretheorem[sibling=theorem]{proposition} -\declaretheorem[sibling=theorem]{corollary} -\declaretheorem[sibling=theorem]{remark} -\declaretheorem[sibling=theorem]{lemma} -\declaretheorem[sibling=theorem]{definition} -\declaretheorem[sibling=theorem]{example} - % We neutralise the Plastex commands \newcommand{\uses}[1]{} \newcommand{\proves}[1]{} diff --git a/blueprint/src/preamble/web.tex b/blueprint/src/preamble/web.tex index dcd75d11..9e38aaf7 100644 --- a/blueprint/src/preamble/web.tex +++ b/blueprint/src/preamble/web.tex @@ -1,11 +1,2 @@ % Those macros are used for the web version of the blueprint. % This file is not meant to be built. Build src/web.tex or src/print.text instead. - -\newtheorem{theorem}{Theorem}[chapter] -\newtheorem{definition}[theorem]{Definition} -\newtheorem{proposition}[theorem]{Proposition} -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{sublemma}[theorem]{Sub-lemma} -\newtheorem{corollary}[theorem]{Corollary} -\newtheorem{remark}[theorem]{Remark} -\newtheorem{example}[theorem]{Example} diff --git a/blueprint/src/print.tex b/blueprint/src/print.tex index 7b20ea0b..9e808bfc 100644 --- a/blueprint/src/print.tex +++ b/blueprint/src/print.tex @@ -1,7 +1,11 @@ +% !TeX root = print.tex % This file makes a printable version of the blueprint +% \documentclass[12pt,reqno,a4paper,dvipsnames]{amsart} \documentclass[a4paper]{report} +% \usepackage[utf8]{inputenc} +\usepackage[dvipsnames]{xcolor} \usepackage[textwidth=14cm]{geometry} \usepackage{xfrac} \usepackage{polyglossia} @@ -9,7 +13,6 @@ \usepackage{amsmath,amssymb} \usepackage{enumitem} -\usepackage{hyperref} \usepackage{tikz-cd} @@ -22,19 +25,18 @@ \setmathfont[range=\intprod]{Asana-Math.otf} \setmathfont[range=\int]{Latin Modern Math} -\usepackage[nameinlink, capitalize]{cleveref} - \usepackage{amsthm} \usepackage{etexcmds} \usepackage{thmtools} \input{preamble/common} \input{preamble/print} +\usepackage[nameinlink, capitalize]{cleveref} -\title{Carleson's Theorem Blueprint} -\author{Terence Tao} +\usepackage[style=trad-alpha]{biblatex} +\ExecuteBibliographyOptions{safeinputenc=true,backref=true,giveninits,useprefix=true,maxnames=5,doi=false,eprint=true,isbn=false,url=false} +\bibliography{bibliography.bib} \begin{document} -\maketitle \input{chapter/main} \end{document} diff --git a/blueprint/src/web.tex b/blueprint/src/web.tex index 17e19a0e..20f501b3 100644 --- a/blueprint/src/web.tex +++ b/blueprint/src/web.tex @@ -4,10 +4,8 @@ \documentclass{report} \usepackage{amsmath,amsfonts,amsthm,amssymb} -\usepackage{hyperref} \usepackage{graphicx} \DeclareGraphicsExtensions{.svg,.png,.jpg} -\usepackage[capitalize]{cleveref} \usepackage[showmore, dep_graph, project=../../]{blueprint} \usepackage{tikz-cd} @@ -15,14 +13,11 @@ \input{preamble/common} \input{preamble/web} +\usepackage[capitalize]{cleveref} \github{https://github.com/fpvandoorn/carleson} \dochome{https://fpvandoorn.github.io/carleson/docs} -\title{Carleson} -\author{} - \begin{document} -\maketitle \input{chapter/main} \end{document} diff --git a/blueprint/tasks.py b/blueprint/tasks.py index 23807328..96260eb9 100644 --- a/blueprint/tasks.py +++ b/blueprint/tasks.py @@ -12,14 +12,16 @@ def print_bp(ctx): cwd = os.getcwd() os.chdir(BP_DIR) - run('mkdir -p print && cd src && xelatex -output-directory=../print print.tex') + os.makedirs("print", exist_ok=True) + run('cd src && xelatex -output-directory=../print print.tex') os.chdir(cwd) @task def bp(ctx): cwd = os.getcwd() os.chdir(BP_DIR) - run('mkdir -p print && cd src && xelatex -output-directory=../print print.tex') + os.makedirs("print", exist_ok=True) + run('cd src && xelatex -output-directory=../print print.tex') run('cd src && xelatex -output-directory=../print print.tex') os.chdir(cwd)