openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /MonotoneShellTestDensity.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Fourier.SchwartzLpTests | |
| import OAI.MeasureTheory.Falconer.Measures.DistributionAbsoluteContinuity | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Shells.ShellKernels | |
| import OAI.MeasureTheory.Falconer.Shells.ShellSummability | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.ComplexOfPositiveFinite | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| namespace CurrentFalconer.MonotoneShell | |
| open MeasureTheory Set Filter PlanarFalconer | |
| open scoped Topology ENNReal FourierTransform SchwartzMap | |
| lemma monotone_shell_test_density | |
| (J : ℕ) (α : ℕ → FiniteMeasure ℝ) (αinf : FiniteMeasure ℝ) | |
| (τ : ℕ → ComplexMeasure ℝ) | |
| (hfinite : ∀ n, IsFiniteMeasure (τ n).variation) | |
| (hmono : Antitone (fun n => (α n : Measure ℝ))) | |
| (hlim : ∀ s, MeasurableSet s → Tendsto (fun N => (α N : Measure ℝ) s) atTop | |
| (𝓝 ((αinf : Measure ℝ) s))) | |
| (hTV : Summable (fun n => complexVariationError (α (n+1)) (τ n))) | |
| (henergy : Summable (fun n => Real.sqrt (complexShellEnergy (τ n) (n+J+1)))) : | |
| ∃ a : Lp (α:=ℝ) ℂ 1 volume, ∃ b : Lp (α:=ℝ) ℂ 2 volume, | |
| ∀ φ : 𝓢(ℝ,ℂ), (∫ x, φ x*(a x+b x)) = ∫ x, φ x ∂(αinf : Measure ℝ) := by | |
| let (n : ℕ) := hfinite n | |
| have (n : ℕ) : IsFiniteMeasure (complexOfPositive (α (n+1))-τ n).variation := | |
| isFiniteMeasure_of_le ((complexOfPositive (α (n+1))).variation+(τ n).variation) | |
| VectorMeasure.variation_sub_le | |
| let e (n : ℕ) := complexBand1 (complexOfPositive (α (n+1))-τ n) (n+J) | |
| let g (n : ℕ) := complexBand2 (τ n) (n+J) | |
| have he : Summable e := by | |
| apply Summable.of_norm | |
| apply (hTV.mul_left (2 * ∫ x, ‖sourceShellKernel 0 x‖)).of_nonneg_of_le | |
| (fun n => norm_nonneg _) (fun n => ?_) | |
| have h := complexBand1_norm (complexOfPositive (α (n+1))-τ n) (n+J) | |
| simpa only [e,complexVariationError,mul_assoc,mul_left_comm,mul_comm] using h | |
| have hg : Summable g := by | |
| apply Summable.of_norm | |
| exact (henergy.mul_left (Real.sqrt ((2*Real.pi)⁻¹))).of_nonneg_of_le | |
| (fun n => norm_nonneg _) (fun n => complexBand2_norm (τ n) (n+J)) | |
| obtain ⟨Z,hZ⟩ := monotone_correction_converges J α αinf hmono hlim | |
| let a := lowTerm J (αinf : Measure ℝ) + ∑' n, e n - Z | |
| let b := ∑' n, g n | |
| refine ⟨a,b,?_⟩ | |
| intro φ | |
| have htZ := ((schwartzLpTest 1 φ).continuous.tendsto Z).comp hZ | |
| have ht := (((schwartzLpTest_sum_tendsto 1 φ e he).const_add | |
| (schwartzLpTest 1 φ (lowTerm J (αinf : Measure ℝ)))).sub htZ).add | |
| (schwartzLpTest_sum_tendsto 2 φ g hg) | |
| have hf (N : ℕ) : | |
| ((schwartzLpTest 1 φ (lowTerm J (αinf : Measure ℝ)) + | |
| ∑ n ∈ Finset.range N, schwartzLpTest 1 φ (e n)) - | |
| schwartzLpTest 1 φ (∑ n ∈ Finset.range N, | |
| bandTerm (n+J) ((α (n+1) : Measure ℝ)-(αinf : Measure ℝ)))) + | |
| ∑ n ∈ Finset.range N, schwartzLpTest 2 φ (g n) = | |
| schwartzLpTest 1 φ (lowTerm (N+J) (αinf : Measure ℝ)) := by | |
| rw [bandTerm_telescope J N,map_add,map_sum,map_sum] | |
| have hr (n : ℕ) : schwartzLpTest 1 φ (e n) + schwartzLpTest 2 φ (g n) - | |
| schwartzLpTest 1 φ (bandTerm (n+J) ((α (n+1) : Measure ℝ)-(αinf : Measure ℝ))) = | |
| schwartzLpTest 1 φ (bandTerm (n+J) (αinf : Measure ℝ)) := by | |
| rw [complex_band_recombine,bandTerm_sub_of_le _ _ _ | |
| (monotone_limit_le α αinf hmono hlim (n+1)),map_sub] | |
| abel | |
| calc | |
| _ = schwartzLpTest 1 φ (lowTerm J (αinf : Measure ℝ)) + | |
| ∑ n ∈ Finset.range N, (schwartzLpTest 1 φ (e n) + schwartzLpTest 2 φ (g n) - | |
| schwartzLpTest 1 φ (bandTerm (n+J) ((α (n+1) : Measure ℝ)-(αinf : Measure ℝ)))) := by | |
| rw [Finset.sum_sub_distrib,Finset.sum_add_distrib]; abel | |
| _ = _ := by simp_rw [hr] | |
| simp only [Function.comp_def] at ht | |
| simp_rw [hf] at ht | |
| have hv := tendsto_nhds_unique ht (lowTerm_test_tendsto J αinf φ) | |
| have hval : schwartzLpTest 1 φ a+schwartzLpTest 2 φ b = ∫ x, φ x ∂(αinf : Measure ℝ) := by | |
| simpa only [a,b,map_sub,map_add] using hv | |
| have ha := (φ.memLp_top volume).integrable_mul (Lp.memLp a) | |
| have hb := (φ.memLp 2 volume).integrable_mul (Lp.memLp b) | |
| change Integrable (fun x => φ x*a x) at ha | |
| change Integrable (fun x => φ x*b x) at hb | |
| rw [←hval,schwartzLpTest_apply,schwartzLpTest_apply] | |
| simp_rw [mul_add] | |
| exact integral_add ha hb | |
| end CurrentFalconer.MonotoneShell | |
| end | |
| end | |
| section | |
| section | |
| namespace CurrentFalconer.MonotoneShell | |
| open MeasureTheory Set Filter | |
| open scoped Topology ENNReal | |
| theorem monotone_measure_shell_criterion | |
| (K : ℕ) (α : ℕ → FiniteMeasure ℝ) (αinf : FiniteMeasure ℝ) | |
| (τ : ℕ → ComplexMeasure ℝ) | |
| (hfinite : ∀ N, IsFiniteMeasure (τ N).variation) | |
| (hmono : ∀ N ≥ K, (α (N+1) : Measure ℝ) ≤ (α N : Measure ℝ)) | |
| (hlim : ∀ s, MeasurableSet s → | |
| Tendsto (fun N => (α N : Measure ℝ) s) atTop (𝓝 ((αinf : Measure ℝ) s))) | |
| (hTV : Summable (fun n => complexVariationError (α (n+K)) (τ (n+K)))) | |
| (henergy : Summable (fun n => Real.sqrt (complexShellEnergy (τ (n+K)) (n+K)))) : | |
| (αinf : Measure ℝ) ≪ volume | |
| := by | |
| let μ (n : ℕ) := α (n+K) | |
| let ν (n : ℕ) := τ (n+1+K) | |
| have hm : Antitone (fun n => (μ n : Measure ℝ)) := by | |
| apply antitone_nat_of_succ_le | |
| intro n | |
| simpa only [μ,Nat.add_assoc,Nat.add_left_comm,Nat.add_comm] using hmono (n+K) (by omega) | |
| have hl : ∀ s, MeasurableSet s → Tendsto (fun n => (μ n : Measure ℝ) s) atTop | |
| (𝓝 ((αinf : Measure ℝ) s)) := by | |
| intro s hs | |
| exact (hlim s hs).comp (tendsto_add_atTop_nat K) | |
| have h1 : Summable (fun n => complexVariationError (μ (n+1)) (ν n)) := by | |
| exact (summable_nat_add_iff (f := fun n => complexVariationError (α (n+K)) (τ (n+K))) 1).mpr hTV | |
| have h2 : Summable (fun n => Real.sqrt (complexShellEnergy (ν n) (n+K+1))) := by | |
| have h := (summable_nat_add_iff (f := fun n => Real.sqrt (complexShellEnergy (τ (n+K)) (n+K))) 1).mpr henergy | |
| simpa only [ν,Nat.add_assoc,Nat.add_left_comm,Nat.add_comm] using h | |
| obtain ⟨a,b,ht⟩ := monotone_shell_test_density K μ αinf ν (fun n => hfinite (n+1+K)) hm hl h1 h2 | |
| apply PlanarFalconer.absolutelyContinuous_of_smooth_test_density αinf (fun x => a x+b x) | |
| · exact ((MeasureTheory.Lp.memLp a).locallyIntegrable (by norm_num)).add | |
| ((MeasureTheory.Lp.memLp b).locallyIntegrable (by norm_num)) | |
| · intro g hg hc | |
| let φ : SchwartzMap ℝ ℂ := (hc.comp_left (map_zero Complex.ofRealCLM)).toSchwartzMap | |
| (Complex.ofRealCLM.contDiff.comp hg) | |
| exact (ht φ).symm | |
| end CurrentFalconer.MonotoneShell | |
| end | |
| end | |
| section | |
| section | |
| namespace CurrentFalconer.MonotoneShell | |
| open MeasureTheory Set Filter PlanarFalconer | |
| open scoped Topology ENNReal | |
| lemma complexVariationError_nonneg (μ : FiniteMeasure ℝ) (ν : ComplexMeasure ℝ) : | |
| 0 ≤ complexVariationError μ ν := ENNReal.toReal_nonneg | |
| lemma shell_sqrt_summable_of_decay (τ : ℕ → ComplexMeasure ℝ) {γ C : ℝ} | |
| (hγ : 0 < γ) (hC : 0 ≤ C) (N₀ : ℕ) | |
| (h : ∀ N ≥ N₀, complexShellEnergy (τ N) N ≤ C*(2:ℝ)^(-γ*(N:ℝ))) : | |
| Summable (fun N => Real.sqrt (complexShellEnergy (τ N) N)) := by | |
| have hs := (summable_dyadic_decay (half_pos hγ)).mul_left (Real.sqrt C) | |
| apply hs.of_norm_bounded_eventually_nat | |
| filter_upwards [eventually_ge_atTop N₀] with N hN | |
| rw [Real.norm_eq_abs,abs_of_nonneg (Real.sqrt_nonneg _)] | |
| apply (sq_le_sq₀ (Real.sqrt_nonneg _) (by positivity)).mp | |
| rw [Real.sq_sqrt (complexShellEnergy_nonneg _ _),mul_pow,Real.sq_sqrt hC, | |
| ←Real.rpow_natCast _ 2,←Real.rpow_mul (by norm_num)] | |
| convert h N hN using 1; congr 2; push_cast; ring | |
| theorem monotone_measure_shell_criterion_of_decay | |
| (K : ℕ) (α : ℕ → FiniteMeasure ℝ) (αinf : FiniteMeasure ℝ) | |
| (τ : ℕ → ComplexMeasure ℝ) | |
| (hfinite : ∀ N, IsFiniteMeasure (τ N).variation) | |
| (hmono : ∀ N ≥ K, (α (N+1) : Measure ℝ) ≤ (α N : Measure ℝ)) | |
| (hlim : ∀ s, MeasurableSet s → Tendsto (fun N => (α N : Measure ℝ) s) atTop | |
| (𝓝 ((αinf : Measure ℝ) s))) | |
| {η γ A B : ℝ} (hη : 0 < η) (hγ : 0 < γ) (hB : 0 ≤ B) | |
| (N₀ : ℕ) (hTV : ∀ N ≥ N₀, complexVariationError (α N) (τ N) ≤ A*(2:ℝ)^(-η*(N:ℝ))) | |
| (henergy : ∀ N ≥ N₀, complexShellEnergy (τ N) N ≤ B*(2:ℝ)^(-γ*(N:ℝ))) : | |
| (αinf : Measure ℝ) ≪ volume := by | |
| have h1 : Summable (fun N => complexVariationError (α N) (τ N)) := by | |
| apply ((summable_dyadic_decay hη).mul_left A).of_norm_bounded_eventually_nat | |
| filter_upwards [eventually_ge_atTop N₀] with N hN | |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (complexVariationError_nonneg _ _)] using hTV N hN | |
| have h2 := shell_sqrt_summable_of_decay τ hγ hB N₀ henergy | |
| exact monotone_measure_shell_criterion K α αinf τ hfinite hmono hlim | |
| ((summable_nat_add_iff (f := fun N => complexVariationError (α N) (τ N)) K).mpr h1) | |
| ((summable_nat_add_iff (f := fun N => Real.sqrt (complexShellEnergy (τ N) N)) K).mpr h2) | |
| end CurrentFalconer.MonotoneShell | |
| end | |
| end | |
| section | |
| namespace CurrentFalconer.AssemblyCompletion | |
| open Set Metric Filter | |
| open scoped Topology ContDiff | |
| lemma shell_cutoff_exists : ∃ Ψ : ℝ → ℝ, ContDiff ℝ ∞ Ψ ∧ | |
| tsupport Ψ⊆Ioo (1/4:ℝ) 4 ∧ (∀ r,0≤Ψ r) ∧ (∀ r∈Icc (1/2:ℝ) 2,1≤Ψ r) := by | |
| let f : ContDiffBump (5/4:ℝ) := ⟨4/5,9/10,by norm_num,by norm_num⟩ | |
| refine ⟨f,f.contDiff,?_,f.nonneg',?_⟩ | |
| · rw [f.tsupport_eq] | |
| intro r hr | |
| change |r-5/4|≤9/10 at hr | |
| obtain ⟨hl,hu⟩ := abs_le.mp hr | |
| constructor <;> linarith | |
| · intro r hr | |
| have hm : r∈closedBall (5/4:ℝ) f.rIn := by | |
| change |r-5/4|≤4/5 | |
| rw [abs_le] | |
| constructor <;> linarith [hr.1,hr.2] | |
| rw [f.one_of_mem_closedBall hm] | |
| lemma direction_cutoff_exists {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] (w₀ : V) (hw₀ : ‖w₀‖=1) : | |
| ∃ ψ : V → ℂ, ContDiff ℝ ∞ ψ ∧ (∀ w, ‖w‖=1 → ‖ψ w‖≤1) ∧ | |
| ∀ w : V, ‖w-w₀‖≤1/100 → | |
| ψ =ᶠ[𝓝 w] (fun _ => 1) ∧ ψ =ᶠ[𝓝 (-w)] (fun _ => 0) := by | |
| let f : ContDiffBump w₀ := ⟨1/50,1/10,by norm_num,by norm_num⟩ | |
| refine ⟨fun w => (f w:ℂ),Complex.ofRealCLM.contDiff.comp f.contDiff,?_,?_⟩ | |
| · intro w _ | |
| simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg f.nonneg] using (f.le_one (x:=w)) | |
| · intro w hw | |
| constructor | |
| · have h : w∈ball w₀ f.rIn := by | |
| rw [mem_ball,dist_eq_norm] | |
| change ‖w-w₀‖<1/50 | |
| linarith | |
| filter_upwards [f.eventuallyEq_one_of_mem_ball h] with z hz | |
| simp only [Pi.one_apply] at hz | |
| simp [hz] | |
| · have hdist : f.rOut<dist (-w) w₀ := by | |
| have hh := norm_add_le (w+w₀) (w₀-w) | |
| have hadd : w+w₀+(w₀-w)= (2:ℝ) • w₀ := by module | |
| rw [hadd,norm_smul,hw₀] at hh | |
| have hn : ‖w₀-w‖=‖w-w₀‖ := norm_sub_rev _ _ | |
| rw [hn] at hh | |
| have he : dist (-w) w₀=‖w+w₀‖ := by rw [dist_eq_norm,show -w-w₀=-(w+w₀) by module,norm_neg] | |
| rw [he] | |
| change (1:ℝ)/10<‖w+w₀‖ | |
| norm_num at hh | |
| linarith | |
| have hev : ∀ᶠ z in 𝓝 (-w), f.rOut<dist z w₀ := | |
| (continuous_id.dist continuous_const).continuousAt.eventually (lt_mem_nhds hdist) | |
| filter_upwards [hev] with z hz | |
| simp [f.zero_of_le_dist hz.le] | |
| end CurrentFalconer.AssemblyCompletion | |
| end | |
| end OAI | |