| import Mathlib |
| import OAI.Analysis.FourierExtension.L2Fourier |
|
|
| namespace OAI |
|
|
| /-! Surface-area measure bounds in graph coordinates. -/ |
|
|
| open MeasureTheory |
| open scoped NNReal ENNReal ContDiff |
| noncomputable section |
| open Filter |
| open scoped Topology ContDiff |
| open MeasureTheory Set |
| open scoped ContDiff FourierTransform InnerProductSpace ENNReal |
| open MeasureTheory Set Filter Metric |
| open scoped ContDiff Topology |
| open Set Filter |
| open scoped ENNReal InnerProductSpace |
| open scoped FourierTransform SchwartzMap ENNReal |
| open scoped ENNReal FourierTransform InnerProductSpace |
| open MeasureTheory Set Filter |
| open scoped ContDiff Topology ENNReal |
|
|
| open MeasureTheory Set |
| open scoped ENNReal NNReal |
| namespace DiagonalExtension.ChartMeasure |
| variable {E Y : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E] [Measure.IsAddHaarMeasure (volume : Measure E)] |
| [NormedAddCommGroup Y] [NormedSpace ℝ Y] [MeasurableSpace Y] [BorelSpace Y] |
|
|
| def dimensionFactor (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E] [Measure.IsAddHaarMeasure (volume : Measure E)] : ℝ≥0 := |
| Measure.addHaarScalarFactor (Measure.hausdorffMeasure (Module.finrank ℝ E) : Measure E) (volume : Measure E) |
|
|
| lemma hausdorff_eq (hd : Module.finrank ℝ E = 2) : |
| (Measure.hausdorffMeasure 2 : Measure E) = dimensionFactor E • volume := by |
| simpa [dimensionFactor,hd] using |
| (Measure.isAddLeftInvariant_eq_smul |
| (Measure.hausdorffMeasure (Module.finrank ℝ E) : Measure E) volume) |
|
|
| omit [NormedSpace ℝ Y] in |
| theorem coordinate_measure_le {Γ : E → Y} {π : Y → E} {K : Set E} {L κ : ℝ≥0} |
| (hd : Module.finrank ℝ E = 2) (hπ : Measurable π) |
| (hleft : ∀ x ∈ K, π (Γ x) = x) (hLip : LipschitzOnWith L Γ K) : |
| Measure.map π (((κ : ℝ≥0∞) • (Measure.hausdorffMeasure 2 : Measure Y)).restrict (Γ '' K)) ≤ |
| ((κ : ℝ≥0∞) * (L : ℝ≥0∞)^2 * (dimensionFactor E : ℝ≥0∞)) • volume.restrict K := by |
| apply Measure.le_iff.mpr |
| intro B hB |
| rw [Measure.map_apply hπ hB,Measure.restrict_apply (hπ hB)] |
| have he : π ⁻¹' B ∩ Γ '' K = Γ '' (B ∩ K) := by |
| ext y |
| constructor |
| · rintro ⟨hy, x,hx,rfl⟩ |
| exact ⟨x,⟨by simpa only [mem_preimage,hleft x hx] using hy,hx⟩,rfl⟩ |
| · rintro ⟨x,⟨hxB,hxK⟩,rfl⟩ |
| exact ⟨by simpa only [mem_preimage,hleft x hxK] using hxB,⟨x,hxK,rfl⟩⟩ |
| rw [he,Measure.smul_apply] |
| have hH := (hLip.mono (show B ∩ K ⊆ K from inter_subset_right)).hausdorffMeasure_image_le |
| (by norm_num : (0 : ℝ) ≤ 2) |
| calc |
| _ ≤ (κ : ℝ≥0∞) * ((L : ℝ≥0∞)^2 * (Measure.hausdorffMeasure 2 : Measure E) (B ∩ K)) := by |
| simpa only [ENNReal.rpow_two, smul_eq_mul] using mul_le_mul_right hH (κ : ℝ≥0∞) |
| _ = _ := by |
| rw [hausdorff_eq hd,Measure.smul_apply,Measure.smul_apply,Measure.restrict_apply hB] |
| simp only [smul_eq_mul,ENNReal.smul_def] |
| ring |
|
|
| end DiagonalExtension.ChartMeasure |
|
|
| namespace DiagonalExtension.ChartMeasure |
| open MeasureTheory Set |
| open scoped ENNReal NNReal |
| variable {E Y : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E] |
| [Measure.IsAddHaarMeasure (volume : Measure E)] |
| [NormedAddCommGroup Y] [NormedSpace ℝ Y] [MeasurableSpace Y] [BorelSpace Y] |
|
|
| omit [NormedSpace ℝ Y] in |
| theorem exists_graph_density {Γ : E → Y} {π : Y → E} {K : Set E} {L κ : ℝ≥0} |
| (hd : Module.finrank ℝ E = 2) (hΓ : Continuous Γ) (hπ : Measurable π) |
| (hleft : ∀ x ∈ K, π (Γ x) = x) (hLip : LipschitzOnWith L Γ K) |
| (hK : IsCompact K) {μ : Measure Y} |
| (hμ : μ ≤ ((κ : ℝ≥0∞) • (Measure.hausdorffMeasure 2 : Measure Y)).restrict (Γ '' K)) |
| {f : Y → ℂ} (hf : MemLp f 2 μ) : |
| ∃ g : E → ℂ, MemLp g 2 volume ∧ HasCompactSupport g ∧ Integrable g volume ∧ |
| eLpNorm g 2 volume ≤ |
| ((κ : ℝ≥0∞) * (L : ℝ≥0∞)^2 * (dimensionFactor E : ℝ≥0∞)) ^ (1/2 : ℝ) * |
| eLpNorm f 2 μ ∧ |
| ∀ v : Y → ℂ, Continuous v → (∃ M : ℝ, ∀ y, ‖v y‖ ≤ M) → |
| (∫ x, g x * v (Γ x)) = ∫ y, f y * v y ∂μ := by |
| classical |
| let C : ℝ≥0∞ := (κ : ℝ≥0∞) * (L : ℝ≥0∞)^2 * (dimensionFactor E : ℝ≥0∞) |
| have hC : C ≠ (⊤ : ℝ≥0∞) := by dsimp [C]; finiteness |
| have hcoord : Measure.map π μ ≤ C • volume := |
| ((Measure.map_mono hμ hπ).trans (coordinate_measure_le hd hπ hleft hLip)).trans |
| (by |
| intro B |
| simp only [Measure.smul_apply, smul_eq_mul] |
| exact mul_le_mul_right (Measure.restrict_le_self B) C) |
| have hsupport : ∀ᵐ y ∂μ, y ∈ Γ '' K := by |
| exact (ae_mono hμ) (ae_restrict_mem (hK.image hΓ).measurableSet) |
| let F := hf.toLp f |
| have hF : (F : Y → ℂ) =ᵐ[μ] f := MemLp.coeFn_toLp hf |
| obtain ⟨u,huInt,huNorm,huSupp,huPair⟩ := |
| BoundedPullback.exists_integrable_density hπ hC hcoord F hK.measurableSet |
| hK.measure_lt_top (by |
| filter_upwards [hsupport] with y hy |
| obtain ⟨x,hx,rfl⟩ := hy |
| simp [hleft x hx,hx]) |
| let g := K.indicator (u : E → ℂ) |
| have hgue : g =ᵐ[volume] u := by |
| filter_upwards [huSupp] with x hx |
| by_cases h : x ∈ K |
| · simp [g,h] |
| · simp [g,h,hx h] |
| have hg : MemLp g 2 volume := (Lp.memLp u).ae_eq hgue.symm |
| have hgc : HasCompactSupport g := hK.of_isClosed_subset (isClosed_tsupport _) |
| (closure_minimal (Set.support_indicator_subset) hK.isClosed) |
| refine ⟨g,hg,hgc,huInt.congr hgue.symm,?_,?_⟩ |
| · rw [eLpNorm_congr_ae hgue, ← Lp.enorm_def] |
| rw [← ofReal_norm] |
| calc |
| _ ≤ ENNReal.ofReal (C.toReal ^ (1 / 2 : ℝ) * ‖F‖) := ENNReal.ofReal_le_ofReal huNorm |
| _ = C ^ (1/2 : ℝ) * ENNReal.ofReal ‖F‖ := by |
| rw [ENNReal.ofReal_mul (Real.rpow_nonneg ENNReal.toReal_nonneg _), |
| ← ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg (by norm_num), ENNReal.ofReal_toReal hC] |
| _ = _ := by rw [ofReal_norm, Lp.enorm_def, eLpNorm_congr_ae hF] |
| · intro v hv hvB |
| obtain ⟨M,hM⟩ := hvB |
| let w := K.indicator (fun x => v (Γ x)) |
| have hw : MemLp w 2 volume := by |
| rw [memLp_indicator_iff_restrict hK.measurableSet] |
| have : IsFiniteMeasure ((volume : Measure E).restrict K) := |
| ⟨by simpa using hK.measure_lt_top⟩ |
| exact MemLp.of_bound (hv.comp hΓ).aestronglyMeasurable M |
| (Filter.Eventually.of_forall (fun x => hM (Γ x))) |
| calc |
| (∫ x, g x * v (Γ x)) = ∫ x, u x * w x := by |
| apply integral_congr_ae |
| filter_upwards [huSupp] with x hx |
| by_cases h : x ∈ K <;> simp [g,w,h,hx] |
| _ = ∫ y, F y * w (π y) ∂μ := huPair w hw |
| _ = ∫ y, f y * v y ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [hF,hsupport] with y hy hs |
| obtain ⟨x,hx,rfl⟩ := hs |
| simp [hy,w,hleft x hx,hx] |
|
|
| end DiagonalExtension.ChartMeasure |
|
|
| end |
|
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| end OAI |
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