| import OAI.Analysis.DiskMaximal.AbsoluteContinuity | |
| namespace OAI | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal Topology ContDiff | |
| open scoped RealInnerProductSpace | |
| open MeasureTheory Set Filter | |
| open scoped Topology ENNReal | |
| open scoped Real RealInnerProductSpace | |
| open MeasureTheory Set Filter | |
| open scoped Topology RealInnerProductSpace ContDiff | |
| open MeasureTheory Set Filter | |
| open scoped Topology Interval NNReal | |
| open MeasureTheory Set Filter | |
| open scoped Topology NNReal Real | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal NNReal Topology | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal Topology | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal NNReal | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal | |
| open Finset | |
| open scoped RealInnerProductSpace | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal NNReal Topology | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal RealInnerProductSpace | |
| open MeasureTheory Set | |
| open scoped ENNReal NNReal Topology | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal NNReal RealInnerProductSpace | |
| open scoped RealInnerProductSpace | |
| open scoped RealInnerProductSpace | |
| open scoped ContDiff | |
| open Finset | |
| open MeasureTheory Set Metric Filter | |
| open scoped ENNReal NNReal RealInnerProductSpace ContDiff Topology | |
| open MeasureTheory MeasureTheory.Measure Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace ContDiff | |
| open MeasureTheory Set Metric Filter Finset | |
| open scoped ENNReal NNReal RealInnerProductSpace ContDiff Topology | |
| open MeasureTheory Set | |
| open scoped RealInnerProductSpace | |
| noncomputable section | |
| namespace CenteredDiskEndpoint | |
| open WeakSobolev | |
| lemma hasGradient_iff_hasWeakGradient {f : Plane → ℝ} {g : Plane → Plane} | |
| (hf : LocallyIntegrable f) (hg : Integrable g) : | |
| HasGradient (μ := volume) f g ↔ HasWeakGradient f g := by | |
| constructor | |
| · intro hw φ hφ hc i | |
| simpa only [testPartial,EuclideanSpace.inner_single_right,RCLike.conj_to_real,one_mul] | |
| using hw φ hφ hc (EuclideanSpace.single i 1) | |
| · intro hw φ hφ hc v | |
| have hv : v=∑ i : Fin 2,v i • EuclideanSpace.single i 1 := by | |
| simpa only [EuclideanSpace.basisFun_repr,EuclideanSpace.basisFun_apply] using | |
| ((EuclideanSpace.basisFun (Fin 2) ℝ).sum_repr v).symm | |
| have hleft i : Integrable (fun x => f x*testPartial φ i x) := | |
| hf.integrable_smul_right_of_hasCompactSupport | |
| ((hφ.continuous_fderiv (by simp)).clm_apply continuous_const) (hc.fderiv_apply ℝ _) | |
| have hright i : Integrable (fun x => g x i*φ x) := | |
| ((EuclideanSpace.proj (𝕜 := ℝ) i).integrable_comp hg).locallyIntegrable.integrable_smul_right_of_hasCompactSupport hφ.continuous hc | |
| have hd x : f x*fderiv ℝ φ x v=∑ i : Fin 2,v i*(f x*testPartial φ i x) := by | |
| conv_lhs => rw [hv] | |
| simp only [_root_.map_sum,map_smul,smul_eq_mul,Finset.mul_sum,testPartial] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| ring | |
| have hi x : ⟪g x,v⟫*φ x=∑ i : Fin 2,v i*(g x i*φ x) := by | |
| conv_lhs => rw [hv] | |
| simp only [inner_sum,real_inner_smul_right,EuclideanSpace.inner_single_right,RCLike.conj_to_real,one_mul,Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| ring | |
| simp_rw [hd,hi] | |
| rw [integral_finsetSum _ (fun i _ => (hleft i).const_mul _), | |
| integral_finsetSum _ (fun i _ => (hright i).const_mul _)] | |
| simp only [integral_const_mul,hw φ hφ hc,Finset.sum_neg_distrib,mul_neg] | |
| /-- The coordinate isometry preserves Euclidean, not max-product, norms. -/ | |
| def complexPlane : ℂ ≃ₗᵢ[ℝ] Plane := Complex.orthonormalBasisOneI.repr | |
| lemma integrable_comp_complexPlane {F : Type*} [NormedAddCommGroup F] {f : Plane → F} : | |
| Integrable (fun z => f (complexPlane z)) ↔ Integrable f := | |
| complexPlane.measurePreserving.integrable_comp_emb complexPlane.toMeasurableEquiv.measurableEmbedding | |
| lemma integrable_comp_complexPlane_symm {F : Type*} [NormedAddCommGroup F] {f : ℂ → F} : | |
| Integrable (fun x => f (complexPlane.symm x)) ↔ Integrable f := | |
| complexPlane.symm.measurePreserving.integrable_comp_emb complexPlane.symm.toMeasurableEquiv.measurableEmbedding | |
| lemma locallyIntegrable_comp_complexPlane {f : Plane → ℝ} : | |
| LocallyIntegrable (fun z => f (complexPlane z)) ↔ LocallyIntegrable f := by | |
| have h := locallyIntegrable_map_homeomorph complexPlane.toHomeomorph (f := f) (μ := volume) | |
| change LocallyIntegrable f (Measure.map complexPlane volume) ↔ LocallyIntegrable (fun z => f (complexPlane z)) at h | |
| rw [complexPlane.measurePreserving.map_eq] at h | |
| exact h.symm | |
| lemma gradientNorm_comp_complexPlane (g : Plane → Plane) : | |
| (∫ z,‖complexPlane.symm (g (complexPlane z))‖)=gradientNorm g := by | |
| simp only [LinearIsometryEquiv.norm_map,gradientNorm] | |
| exact complexPlane.measurePreserving.integral_comp complexPlane.toMeasurableEquiv.measurableEmbedding (fun x => ‖g x‖) | |
| lemma gradientNorm_comp_complexPlane_symm (G : ℂ → ℂ) : | |
| gradientNorm (fun x => complexPlane (G (complexPlane.symm x)))=∫ z,‖G z‖ := by | |
| simp only [LinearIsometryEquiv.norm_map,gradientNorm] | |
| exact complexPlane.symm.measurePreserving.integral_comp complexPlane.symm.toMeasurableEquiv.measurableEmbedding (fun z => ‖G z‖) | |
| end CenteredDiskEndpoint | |
| namespace CenteredDiskEndpoint | |
| open WeakSobolev DiskMaximalTools | |
| /-- Equality of the actual all-real-radius operators, including the normalization | |
| and absolute values. This bridge has no regularity assumption on the input. -/ | |
| lemma maximalComplex_comp_complexPlane (f : Plane → ℝ) (z : ℂ) : | |
| maximalComplex (fun w => f (complexPlane w)) z=maximal f (complexPlane z) := by | |
| rw [maximalComplex_eq_all_radii,maximal] | |
| apply iSup_congr | |
| intro r | |
| apply iSup_congr | |
| intro hr | |
| rw [absoluteDiskAverage,absoluteMeasure,withDensity_apply _ measurableSet_ball, | |
| volume_disk_real z hr.le] | |
| congr 1 | |
| have h := complexPlane.measurePreserving.setLIntegral_comp_emb | |
| complexPlane.toMeasurableEquiv.measurableEmbedding (fun x => ENNReal.ofReal |f x|) (Metric.ball z r) | |
| rw [complexPlane.image_ball] at h | |
| exact h | |
| lemma maximalComplexReal_comp_complexPlane (f : Plane → ℝ) : | |
| DiskMaximalTools.maximalComplexReal (fun w => f (complexPlane w)) = fun z => maximalReal f (complexPlane z) := by | |
| funext z | |
| exact congrArg ENNReal.toReal (maximalComplex_comp_complexPlane f z) | |
| theorem main_endpoint : EndpointStatement := by | |
| obtain ⟨C,hC,hendpoint⟩ := SobolevBand.complex_endpoint | |
| refine ⟨C,hC.le,?_⟩ | |
| intro f g hf hg hw | |
| let f' := fun z => f (complexPlane z) | |
| let g' := fun z => complexPlane.symm (g (complexPlane z)) | |
| have hf' : Integrable f' := integrable_comp_complexPlane.mpr hf | |
| have hg0 : Integrable (fun z => g (complexPlane z)) := integrable_comp_complexPlane.mpr hg | |
| have hg' : Integrable g' := complexPlane.symm.toContinuousLinearEquiv.toContinuousLinearMap.integrable_comp hg0 | |
| have hw0 : HasGradient (μ := volume) f g := (hasGradient_iff_hasWeakGradient hf.locallyIntegrable hg).mpr hw | |
| have hw' : HasGradient (μ := volume) f' g' := hw0.comp_isometry complexPlane | |
| obtain ⟨hM',G',hG'i,hwG',hbound⟩ := hendpoint f' g' hf' hg' hw' | |
| have he : maximalComplexReal f'=fun z => maximalReal f (complexPlane z) := maximalComplexReal_comp_complexPlane f | |
| rw [he] at hM' | |
| have hM : LocallyIntegrable (maximalReal f) := locallyIntegrable_comp_complexPlane.mp hM' | |
| let G := fun x => complexPlane (G' (complexPlane.symm x)) | |
| have hG0 : Integrable (fun x => G' (complexPlane.symm x)) := integrable_comp_complexPlane_symm.mpr hG'i | |
| have hG : Integrable G := complexPlane.toContinuousLinearEquiv.toContinuousLinearMap.integrable_comp hG0 | |
| have hwG0 := hwG'.comp_isometry complexPlane.symm | |
| have hwG : HasWeakGradient (maximalReal f) G := by | |
| apply (hasGradient_iff_hasWeakGradient hM hG).mp | |
| simpa only [he,complexPlane.apply_symm_apply,LinearIsometryEquiv.symm_symm] using hwG0 | |
| refine ⟨ae_maximal_ne_top hf,hM,inW11Loc_of_gradient hM hG hwG,G,hG,hwG,?_⟩ | |
| rw [show gradientNorm G=∫ z,‖G' z‖ from gradientNorm_comp_complexPlane_symm G'] | |
| exact hbound.trans_eq (congrArg (fun v => C*v) (gradientNorm_comp_complexPlane g)) | |
| end CenteredDiskEndpoint | |
| end | |
| end OAI | |