File size: 3,139 Bytes
cead56f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | 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
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