| import OAI.Geometry.RCD.Coordinates.ComparableRadiusNormalization |
| import OAI.Geometry.RCD.Coordinates.NormalizedOuterDirectionError |
| import OAI.Geometry.RCD.Sobolev.HarmonicL1GradientBound |
| import OAI.Geometry.RCD.Harmonic.DirectionalToGramError |
| import OAI.Geometry.RCD.Harmonic.ColumnRepresentatives |
|
|
| namespace OAI |
|
|
| /-! # Full transformed splitting estimates at comparable radii |
|
|
| This is the analytic one-step part of transformation. The all-scale |
| geometric propagation is not assumed or asserted here. |
| -/ |
|
|
| open Set Metric MeasureTheory |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
|
|
| theorem NoncollapsedRCD.interior_radius_splitting {K : ℝ} {n : ℕ} |
| (hX : NoncollapsedRCD K n X) [(hausdorff n X).Regular] (hn : 2 ≤ n) |
| (τ : ℝ) (hτ : 0 < τ) (hτ1 : τ ≤ 1) (β : ℝ) (hβ : 0 < β) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ (p : X) (R : ℝ), 0 < R → -1 ≤ K*(2*R)^2 → |
| ∀ (u : Fin n → SobolevSpace (hausdorff n X) hX.quadratic) (v : X → Euclidean n), |
| FirstOrderSplittingOn (hausdorff n X) hX.quadratic p (2*R) δ u v → |
| ∀ t : ℝ, τ*R ≤ t → t ≤ R → ∃ A : Euclidean n ≃L[ℝ] Euclidean n, |
| FirstOrderSplittingOn (hausdorff n X) hX.quadratic p t β |
| (transformedColumn (hausdorff n X) hX.quadratic u A) |
| (fun x => (A : Euclidean n →L[ℝ] Euclidean n).adjoint (v x)) ∧ |
| IsExactMeanNormalization (hausdorff n X) hX.quadratic |
| (sobolevColumnMap (hausdorff n X) hX.quadratic u) (ball p t) measurableSet_ball A := by |
| obtain ⟨C,hC,hnormalize⟩ := hX.interior_radius_normalization hn τ hτ hτ1 |
| obtain ⟨θ,hθ,hsharp⟩ := hX.harmonic_l1_gradient_bound hn β (8*(n : ℝ)^2) hβ (by positivity) |
| let D := 8*(n : ℝ)^2*C+1 |
| have ha : 0 ≤ (n : ℝ)^2*C := mul_nonneg (sq_nonneg _) (zero_le_one.trans hC) |
| have hD : 0 < D := by dsimp only [D]; nlinarith |
| let δ := min 1 (min ((1/2)/D) (min (β/D) (θ/D))) |
| have hδ : 0 < δ := lt_min zero_lt_one |
| (lt_min (div_pos (by norm_num) hD) (lt_min (div_pos hβ hD) (div_pos hθ hD))) |
| have hδ1 : δ ≤ 1 := min_le_left _ _ |
| have hhalf : D*δ ≤ 1/2 := by |
| have hh : δ ≤ (1/2)/D := (min_le_right _ _).trans (min_le_left _ _) |
| nlinarith [(le_div_iff₀ hD).mp hh] |
| have hbet : D*δ ≤ β := by |
| have hh : δ ≤ β/D := (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)) |
| nlinarith [(le_div_iff₀ hD).mp hh] |
| have htheta : D*δ ≤ θ := by |
| have hh : δ ≤ θ/D := (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _)) |
| nlinarith [(le_div_iff₀ hD).mp hh] |
| have hsmall : (n : ℝ)^2*(C*δ) ≤ 1/2 := by dsimp only [D] at hhalf; nlinarith |
| have hgramsmall : 8*(n : ℝ)^2*(C*δ) ≤ β := by dsimp only [D] at hbet; nlinarith |
| have herrorsmall : 4*(n : ℝ)^2*(C*δ) ≤ θ := by |
| dsimp only [D] at htheta |
| nlinarith [mul_nonneg ha hδ.le] |
| refine ⟨δ,hδ,?_⟩ |
| intro p R hR hcurv u v hfirst t ht htR |
| let μ := hausdorff n X |
| let U := sobolevColumnMap μ hX.quadratic u |
| have ht0 : 0 < t := lt_of_lt_of_le (by positivity : 0 < τ*R) ht |
| have htSub : ball p t ⊆ ball p (2*R) := ball_subset_ball (by linarith) |
| have hRSub : ball p t ⊆ ball p R := ball_subset_ball htR |
| have hz : μ (ball p t) ≠ 0 := (hX.fullSupport _ _ ht0).ne' |
| have hf : μ (ball p t) ≠ ⊤ := (hX.finiteOnBounded _ isBounded_ball).ne |
| have hΩz : μ (ball p (2*R)) ≠ 0 := (hX.fullSupport _ _ (by positivity)).ne' |
| have hΩf : μ (ball p (2*R)) ≠ ⊤ := (hX.finiteOnBounded _ isBounded_ball).ne |
| obtain ⟨hinner,A,hA,hnorm,hdir⟩ := hnormalize p R δ hR hcurv hδ.le hsmall u v hfirst t ht htR |
| have houter (z : Euclidean n) : (μ.real (ball p (2*R)))⁻¹*(∫ x in ball p (2*R), |
| |localGamma μ hX.quadratic (U (A z)) (U (A z)) x-‖z‖^2| ∂μ) |
| ≤ 4*(n : ℝ)^2*(C*δ)*‖z‖^2 := by |
| have he := normalized_outer_direction_error μ hX.quadratic u (ball p t) (ball p (2*R)) |
| measurableSet_ball hz hf hΩz hΩf hδ.le (mul_nonneg (zero_le_one.trans hC) hδ.le) |
| hsmall hinner.gram hfirst.gram A hA z |
| have hcoef : 2*(n : ℝ)^2*(δ+C*δ) ≤ 4*(n : ℝ)^2*(C*δ) := by |
| nlinarith [mul_nonneg (sq_nonneg (n : ℝ)) (mul_nonneg (sub_nonneg.mpr hC) hδ.le)] |
| exact he.trans (mul_le_mul_of_nonneg_right hcoef (sq_nonneg _)) |
| have hg (i : Fin n) : ∀ᵐ x ∂μ.restrict (ball p R), |
| localGamma μ hX.quadratic (transformedColumn μ hX.quadratic u A i) |
| (transformedColumn μ hX.quadratic u A i) x ≤ 1+β := by |
| let ei : Euclidean n := EuclideanSpace.single i 1 |
| have hAi : ‖A ei‖^2 ≤ 2 := by |
| have he : ‖A ei‖ ≤ ‖(A : Euclidean n →L[ℝ] Euclidean n)‖ := by |
| simpa only [ei,PiLp.norm_single,norm_one,mul_one,ContinuousLinearEquiv.coe_coe] using |
| (A : Euclidean n →L[ℝ] Euclidean n).le_opNorm ei |
| exact (pow_le_pow_left₀ (norm_nonneg _) he 2).trans hnorm |
| have hcoarse := hfirst.directional_gradient μ hX.quadratic (A ei) |
| have hcoef : ((n : ℝ)*(1+δ))^2 ≤ 4*(n : ℝ)^2 := |
| splittingDirectionBound_sq_le hδ.le hδ1 |
| have hb : ∀ᵐ x ∂μ.restrict (ball p (2*R)), |
| localGamma μ hX.quadratic (U (A ei)) (U (A ei)) x ≤ 8*(n : ℝ)^2 := by |
| filter_upwards [hcoarse] with x hx |
| have hm := mul_le_mul hcoef hAi (sq_nonneg _) (by positivity : 0 ≤ 4*(n : ℝ)^2) |
| nlinarith |
| have he : (μ.real (ball p (2*R)))⁻¹*(∫ x in ball p (2*R), |
| |localGamma μ hX.quadratic (U (A ei)) (U (A ei)) x-1| ∂μ) ≤ θ := by |
| have hh := houter ei |
| simp only [ei,PiLp.norm_single,norm_one,one_pow,mul_one] at hh |
| exact hh.trans herrorsmall |
| exact hsharp p R hR hcurv (U (A ei)) |
| (IsSobolevHarmonicOn.linearCombination μ hX.quadratic hfirst.harmonic (A ei)) hb he |
| have hgrams : ∀ i j : Fin n, (μ.real (ball p t))⁻¹*∫ x in ball p t, |
| |localGamma μ hX.quadratic (transformedColumn μ hX.quadratic u A i) |
| (transformedColumn μ hX.quadratic u A j) x-(if i=j then 1 else 0)| ∂μ ≤ β := by |
| have hh := directional_error_implies_gram μ hX.quadratic (transformedColumn μ hX.quadratic u A) |
| (ball p t) hf (show 0 ≤ 4*(n : ℝ)^2*(C*δ) by positivity) (by |
| intro z |
| rw [sobolevColumnMap_transformed] |
| exact hdir z) |
| intro i j |
| exact (hh i j).trans (by nlinarith only [hgramsmall]) |
| refine ⟨A,⟨?_,?_,?_,?_,hgrams⟩,hA⟩ |
| · intro i |
| have he := sobolevColumnMap_representative μ hX.quadratic u (ball p t) v |
| (fun j => ae_restrict_of_ae_restrict_of_subset htSub (hfirst.representative j)) |
| (A (EuclideanSpace.single i 1)) |
| filter_upwards [he] with x hx |
| change SobolevSpace.toLp μ hX.quadratic (U (A (EuclideanSpace.single i 1))) x = _ |
| rw [hx] |
| have ha := ContinuousLinearMap.adjoint_inner_left |
| (A : Euclidean n →L[ℝ] Euclidean n) (EuclideanSpace.single i 1) (v x) |
| simpa only [real_inner_comm,EuclideanSpace.inner_single_right,starRingEnd_apply, |
| star_trivial,one_mul,ContinuousLinearEquiv.coe_coe] using ha.symm |
| · exact (A : Euclidean n →L[ℝ] Euclidean n).adjoint.continuous.comp_continuousOn |
| (hfirst.continuous.mono htSub) |
| · exact transformedColumn_harmonic μ hX.quadratic u A (ball p t) |
| (fun i => (hfirst.harmonic i).mono μ hX.quadratic htSub) |
| · intro i |
| filter_upwards [ae_restrict_of_ae_restrict_of_subset hRSub (hg i)] with x hx |
| exact hx.trans (by nlinarith : 1+β ≤ (1+β)^2) |
|
|
| theorem NoncollapsedRCD.comparable_radius_splitting {K : ℝ} {n : ℕ} |
| (hX : NoncollapsedRCD K n X) [(hausdorff n X).Regular] (hn : 2 ≤ n) |
| (β : ℝ) (hβ : 0 < β) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ (p : X) (R : ℝ), 0 < R → -1 ≤ K*(2*R)^2 → |
| ∀ (u : Fin n → SobolevSpace (hausdorff n X) hX.quadratic) (v : X → Euclidean n), |
| FirstOrderSplittingOn (hausdorff n X) hX.quadratic p (2*R) δ u v → |
| ∀ t : ℝ, R/2 ≤ t → t ≤ R → ∃ A : Euclidean n ≃L[ℝ] Euclidean n, |
| FirstOrderSplittingOn (hausdorff n X) hX.quadratic p t β |
| (transformedColumn (hausdorff n X) hX.quadratic u A) |
| (fun x => (A : Euclidean n →L[ℝ] Euclidean n).adjoint (v x)) ∧ |
| IsExactMeanNormalization (hausdorff n X) hX.quadratic |
| (sobolevColumnMap (hausdorff n X) hX.quadratic u) (ball p t) measurableSet_ball A := by |
| simpa only [show ∀ R : ℝ, (1/2)*R=R/2 by intro R; ring] using |
| hX.interior_radius_splitting hn (1/2) (by norm_num) (by norm_num) β hβ |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
