forgot to add file

Carleson.lean

@@ -6,6 +6,7 @@ import Carleson.DoublingMeasure
 import Carleson.FinitaryCarleson
 import Carleson.ForestOperator
 import Carleson.GridStructure
+import Carleson.HardyLittlewood
 import Carleson.HolderVanDerCorput
 import Carleson.MetricCarleson
 import Carleson.Proposition2

Carleson/HardyLittlewood.lean

@@ -0,0 +1,81 @@
+import Mathlib.Analysis.NormedSpace.FiniteDimension
+import Mathlib.Analysis.SpecialFunctions.Log.Base
+import Mathlib.MeasureTheory.Integral.Average
+import Mathlib.MeasureTheory.Measure.Haar.Basic
+import Mathlib.MeasureTheory.Measure.Doubling
+import Mathlib.MeasureTheory.Covering.VitaliFamily
+
+open MeasureTheory Metric Bornology Set
+open scoped NNReal ENNReal
+noncomputable section
+
+#check VitaliFamily
+
+variable {X E} [PseudoMetricSpace X] [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E]
+  {μ : Measure X} {f : X → E} {x : X} {𝓑 : Finset (X × ℝ)}
+
+/-- The Hardy-Littlewood maximal function w.r.t. a collection of balls 𝓑. -/
+def maximalFunction (μ : Measure X) (𝓑 : Set (X × ℝ)) (p : ℝ) (u : X → E) (x : X) : ℝ≥0∞ :=
+  (⨆ z ∈ 𝓑, (ball z.1 z.2).indicator (x := x)
+  fun _ ↦ ⨍⁻ y, ‖u y‖₊ ∂μ.restrict (ball z.1 z.2)) ^ p⁻¹
+
+-- old
+-- /-- Hardy-Littlewood maximal function -/
+-- def maximalFunction (μ : Measure X) (f : X → E) (x : X) : ℝ :=
+--   ⨆ (x' : X) (δ : ℝ) (_hx : x ∈ ball x' δ),
+--   ⨍⁻ y, ‖f y‖₊ ∂μ.restrict (ball x' δ) |>.toReal
+
+/-! The following results probably require a doubling measure,
+and maybe some properties from `ProofData`.
+They are the statements from the blueprint.
+We probably want a more general version first. -/
+
+theorem measure_biUnion_le_lintegral {a : ℝ} (ha : 4 ≤ a) {l : ℝ≥0} (hl : 0 < l)
+    {u : X → ℝ≥0} (hu : AEStronglyMeasurable u μ)
+    (h2u : ∀ z ∈ 𝓑, l * μ (ball z.1 z.2) ≤ ∫⁻ x in ball z.1 z.2, u x ∂μ)
+    :
+    l * μ (⋃ z ∈ 𝓑, ball z.1 z.2) ≤ 2 ^ (2 * a) * ∫⁻ x, u x ∂μ  := by
+  sorry
+
+theorem maximalFunction_lt_top {a : ℝ} (ha : 4 ≤ a) {l : ℝ≥0} (hl : 0 < l) {p₁ p₂ : ℝ≥0}
+    (hp₁ : 1 ≤ p₁) (hp₁₂ : p₁ < p₂)
+    {u : X → E} (hu : Memℒp u p₂ μ) {x : X} :
+    maximalFunction μ 𝓑 p₁ u x < ∞ := by
+  sorry
+
+theorem snorm_maximalFunction {a : ℝ} (ha : 4 ≤ a) {l : ℝ≥0} (hl : 0 < l) {p₁ p₂ : ℝ≥0}
+    (hp₁ : 1 ≤ p₁) (hp₁₂ : p₁ < p₂)
+    {u : X → E} (hu : AEStronglyMeasurable u μ) :
+    snorm (maximalFunction μ 𝓑 p₁ u · |>.toReal) p₂ μ ≤
+    2 ^ (2 * a) * p₂ / (p₂ - p₁) * snorm u p₂ μ := by
+  sorry
+
+theorem Memℒp.maximalFunction {a : ℝ} (ha : 4 ≤ a) {l : ℝ≥0} (hl : 0 < l) {p₁ p₂ : ℝ≥0}
+    (hp₁ : 1 ≤ p₁) (hp₁₂ : p₁ < p₂)
+    {u : X → E} (hu : Memℒp u p₂ μ) :
+    Memℒp (maximalFunction μ 𝓑 p₁ u · |>.toReal) p₂ μ := by
+  sorry
+
+/-- The transformation M characterized in Proposition 2.0.6. -/
+def M (u : X → ℂ) (x : X) : ℝ≥0∞ := sorry
+
+theorem M_lt_top {a : ℝ} (ha : 4 ≤ a) {l : ℝ≥0} (hl : 0 < l) {p₁ p₂ : ℝ≥0}
+    (hp₁ : 1 ≤ p₁) (hp₁₂ : p₁ < p₂)
+    {u : X → ℂ} (hu : AEStronglyMeasurable u μ) (hu : IsBounded (range u))
+    {x : X} : M u x < ∞ := by
+  sorry
+
+theorem laverage_le_M {a : ℝ} (ha : 4 ≤ a) {l : ℝ≥0} (hl : 0 < l)
+    {u : X → ℂ} (hu : AEStronglyMeasurable u μ) (hu : IsBounded (range u))
+    {z x : X} {r : ℝ} :
+     ⨍⁻ y, ‖u y‖₊ ∂μ.restrict (ball z r) ≤ M u x := by
+  sorry
+
+theorem snorm_M_le {a : ℝ} (ha : 4 ≤ a) {l : ℝ≥0} (hl : 0 < l){p₁ p₂ : ℝ≥0}
+    (hp₁ : 1 ≤ p₁) (hp₁₂ : p₁ < p₂)
+    {u : X → ℂ} (hu : AEStronglyMeasurable u μ) (hu : IsBounded (range u))
+    {z x : X} {r : ℝ} :
+    snorm (fun x ↦ (M (fun x ↦ u x ^ (p₁ : ℂ)) x).toReal ^ (p₁⁻¹ : ℝ)) p₂ μ ≤
+    2 ^ (4 * a)  * p₂ / (p₂ - p₁) * snorm u p₂ μ := by
+  sorry