| import OAI.Geometry.RCD.Sobolev.CenteredHarmonicEnergy | |
| import OAI.Geometry.RCD.Sobolev.ContinuousLocalGradientBound | |
| import OAI.Geometry.RCD.Harmonic.FirstOrderLocalization | |
| namespace OAI | |
| /-! # Bounded centered harmonic components of a first-order frame -/ | |
| open Set Metric MeasureTheory | |
| open scoped NNReal | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] | |
| theorem NoncollapsedRCD.centered_first_order_frame {K : ℝ} {n : ℕ} | |
| (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n) {p : X} {r ε : ℝ} | |
| (hr : 0 < r) (hε : 0 ≤ ε) (hε1 : ε ≤ 1) | |
| (u : Fin n → SobolevSpace (hausdorff n X) hX.quadratic) (v : X → Euclidean n) | |
| (hsplit : FirstOrderSplittingOn (hausdorff n X) hX.quadratic p (80*r) ε u v) : | |
| ∃ w : Fin n → SobolevSpace (hausdorff n X) hX.quadratic, | |
| (∀ i, IsSobolevHarmonicOn (hausdorff n X) hX.quadratic (ball p (5*r)) (w i)) ∧ | |
| (∀ i, SobolevSpace.toLp (hausdorff n X) hX.quadratic (w i) =ᵐ[ | |
| (hausdorff n X).restrict (ball p (5*r))] fun x => (r⁻¹ • (v x-v p)) i) ∧ | |
| (∀ i, harmonicTargetNorm hX p (5*r) (w i) ≤ 10) ∧ | |
| ∀ i j, (fun x => r^2*localGamma (hausdorff n X) hX.quadratic (w i) (w j) x) =ᵐ[ | |
| (hausdorff n X).restrict (ball p (5*r))] | |
| localGamma (hausdorff n X) hX.quadratic (u i) (u j) := by | |
| let μ := hausdorff n X | |
| have hs40 : ball p (8*(5*r)) ⊆ ball p (80*r) := ball_subset_ball (by linarith) | |
| have hs5 : ball p (5*r) ⊆ ball p (80*r) := ball_subset_ball (by linarith) | |
| have hex (i : Fin n) := hX.centered_harmonic_direction_with_energy hn p | |
| (show 0<5*r by positivity) hr (by norm_num : (0:ℝ)≤10) (u i) | |
| (fun x => v x i) (v p i) ((hsplit.harmonic i).mono μ hX.quadratic hs5) | |
| (ae_restrict_of_ae_restrict_of_subset hs5 (hsplit.representative i)) | |
| have hb (i : Fin n) : ∀ x ∈ ball p (5*r), |v x i-v p i| ≤ 10*r := by | |
| have hg : ∀ᵐ x ∂μ.restrict (ball p (8*(5*r))), | |
| localGamma μ hX.quadratic (u i) (u i) x ≤ ((2 : ℝ≥0) : ℝ)^2 := by | |
| filter_upwards [ae_restrict_of_ae_restrict_of_subset hs40 (hsplit.gradient i)] with x hx | |
| norm_num only [NNReal.coe_ofNat] at * | |
| nlinarith | |
| have hv : ContinuousOn (fun x => v x i) (ball p (8*(5*r))) := | |
| (EuclideanSpace.proj i).continuous.comp_continuousOn (hsplit.continuous.mono hs40) | |
| have hbound := hX.continuous_local_gradient_bound hn (u i) (fun x => v x i) p | |
| (show 0<5*r by positivity) 2 hv | |
| (ae_restrict_of_ae_restrict_of_subset hs40 (hsplit.representative i)) hg | |
| intro x hx | |
| have hh := hbound x hx | |
| norm_num only [NNReal.coe_ofNat] at hh | |
| have hd : dist x p < 5*r := hx | |
| linarith | |
| choose w hw hrep hnorm he using (fun i => hex i (hb i)) | |
| refine ⟨w,hw,?_,hnorm,?_⟩ | |
| · intro i | |
| filter_upwards [hrep i] with x hx | |
| change _ = r⁻¹*(v x i-v p i) | |
| simpa only [div_eq_inv_mul] using hx | |
| · intro i j | |
| exact rescaled_local_pairing_of_centering μ hX.quadratic (ball p (5*r)) hr.ne' | |
| (u i) (u j) (w i) (w j) (he i) (he j) | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |