| import OAI.Geometry.RCD.Harmonic.MeasuredRadialL2 | |
| import OAI.Geometry.RCD.Sobolev.GammaL2Bound | |
| namespace OAI | |
| /-! # Compact radial weights agree on the gradients of supported recoveries -/ | |
| open Set Metric Filter MeasureTheory | |
| open scoped Topology BoundedContinuousFunction | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] | |
| theorem compact_radial_pullback (μ : Measure X) (hQ : QuadraticCheegerEnergy μ) | |
| {n : ℕ} {S c T : ℝ} (hS : 0<S) (hcT : c<T) (hTS : T<S) | |
| (p : ℕ → X) (r ε : ℕ → ℝ) (V : ℕ → X → Euclidean n) | |
| (hr : ∀ k, 0<r k) (hε : Tendsto ε atTop (𝓝 0)) | |
| (hV : ∀ k, IsMeasuredEuclideanApproximation μ n (p k) (r k) S (ε k) (V k)) | |
| (u w : ℕ → SobolevSpace μ hQ) | |
| (hz : ∀ k, ∀ᵐ x ∂μ, x∉ball (p k) (c*r k) → SobolevSpace.toLp μ hQ (u k) x=0) | |
| (F : Euclidean n →ᵇ ℝ) (hF : EqOn F (fun z => ‖z‖^2/2) (closedBall 0 T)) : | |
| ∀ᶠ k in atTop, (fun x => (‖V k x‖^2/2)*localGamma μ hQ (u k) (w k) x)=ᵐ[μ] | |
| (fun x => F (V k x)*localGamma μ hQ (u k) (w k) x) := by | |
| filter_upwards [hε.eventually (gt_mem_nhds (show 0<(T-c)/2 by linarith))] with k hk | |
| filter_upwards [hz k,localGamma_zero_on_level μ hQ (u k) (w k) 0] with x hx hγ | |
| by_cases hxc : x∈ball (p k) (c*r k) | |
| · have hxs : dist x (p k)≤S*r k := | |
| hxc.le.trans (mul_le_mul_of_nonneg_right (hcT.trans hTS).le (hr k).le) | |
| have hd := (abs_le.mp (pointed_radial_distance (hr k) hS.le (hV k).metric hxs)).2 | |
| have hdc : dist x (p k)/r k<c := (div_lt_iff₀ (hr k)).mpr hxc | |
| have hnorm : V k x∈closedBall (0:Euclidean n) T := by | |
| rw [mem_closedBall,dist_zero_right] | |
| linarith | |
| rw [hF hnorm] | |
| · rw [hγ (hx hxc),mul_zero,mul_zero] | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |