diff --git a/blueprint/src/chapter/main.tex b/blueprint/src/chapter/main.tex index 92c6158b..0b2e8d3e 100644 --- a/blueprint/src/chapter/main.tex +++ b/blueprint/src/chapter/main.tex @@ -3337,7 +3337,7 @@ \section{The density arguments}\label{sec-TT*-T*T} We have that \begin{equation}\label{eqttt9} \left|\int \overline{g(x)} \sum_{\fp \in \mathfrak{A}} T_{\fp} f(x)\, d\mu(x)\right|\le - 2^{111a^2}({q}-1)^{-1} \dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12} \|f\|_2\|g\|_2\, . + 2^{111a^3}({q}-1)^{-1} \dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12} \|f\|_2\|g\|_2\, . \end{equation} \end{lemma} \begin{proof} @@ -3363,7 +3363,7 @@ \section{The density arguments}\label{sec-TT*-T*T} \end{equation} We have with \Cref{Hardy-Littlewood} \begin{equation} -\left\|M_{\mathcal{B}, \frac {2q}{3q-2}} f\right\|_2\le 2^{2a}(3\tilde{q}-2)(2\tilde{q}-2)^{-1}\|f\|_2\, . +\left\|M_{\mathcal{B}, \frac {2{\tilde{q}}}{3{\tilde{q}}-2}} f\right\|_2\le 2^{2a}(3\tilde{q}-2)(2\tilde{q}-2)^{-1}\|f\|_2\, . \end{equation} Using $1<\tilde{q}\le 2$ estimates the last display by \begin{equation}\label{eqttt2} @@ -3378,12 +3378,12 @@ \section{The density arguments}\label{sec-TT*-T*T} \le \|g\|_2 \Big\| \sum_{\fp \in \mathfrak{A}} T_{\fp} f \Big\|_2 \end{equation} \begin{equation} - \le 2^{107a^2}\|g\|_2 \| M_{\mathcal{B}}f \|_2 + \le 2^{107a^3}\|g\|_2 \| M_{\mathcal{B}}f \|_2 \end{equation} With \eqref{eqttt1} and \eqref{eqttt2} we can estimate the last display by \begin{equation} - \le 2^{107a^2+2a+2}(\tilde{q}-1)^{-1} \|g\|_2 \|f\|_2\dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12} + \le 2^{107a^3+2a+2}(\tilde{q}-1)^{-1} \|g\|_2 \|f\|_2\dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12} \end{equation} Using $a\ge 4$ and $(\tilde q - 1)^{-1} = (q+1)/(q-1) \le 3(q-1)^{-1}$