File size: 3,445 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 67 68 69 | import OAI.Geometry.RCD.Coordinates.RegularDirichletCore
import OAI.Geometry.RCD.Sobolev.WeakLaplacianProduct
import OAI.Geometry.RCD.Sobolev.GoodLaplacianCutoff
namespace OAI
/-! # Compact regular localization retaining the full L² source
A bounded Lipschitz function with weak L² Laplacian is multiplied by an
actual good cutoff. On its plateau both the function and source agree
with the originals. No constant-source or source-gradient premise is used.
-/
open Set Filter Metric MeasureTheory
open scoped NNReal
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] (μ : Measure X) [IsFiniteMeasureOnCompacts μ] (hQ : QuadraticCheegerEnergy μ)
theorem bounded_regular_localization (u : SobolevSpace μ hQ) (d : Lp ℝ 2 μ)
(hd : IsWeakLaplacian μ hQ u d) (v : X → ℝ) {L : ℝ≥0}
(hv : LipschitzWith L v) (hu : SobolevSpace.toLp μ hQ u =ᵐ[μ] v)
(A : ℝ≥0) (hA : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ hQ u x|≤A)
{p : X} {r C : ℝ} (χ : GoodLaplacianCutoff μ hQ p r C) :
∃ (F : compactLaplacianCore μ hQ univ) (q : Lp ℝ 2 μ),
IsWeakLaplacian μ hQ F.val q ∧
SobolevSpace.toLp μ hQ F.val =ᵐ[μ.restrict (ball p r)] SobolevSpace.toLp μ hQ u ∧
q =ᵐ[μ.restrict (ball p r)] d := by
have hχb : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ hQ χ.sobolev x|≤(1:ℝ≥0) := by
filter_upwards [χ.representative] with x hx
rw [hx,abs_of_nonneg (χ.bounds x).1]
exact (χ.bounds x).2
have hχg := SobolevSpace.minimalSlope_le_lipschitz μ hQ χ.sobolev χ.lipschitz χ.representative
have hΓ : ∀ᵐ x ∂μ, localGamma μ hQ χ.sobolev χ.sobolev x≤((C/r).toNNReal:ℝ)^2 := by
filter_upwards [localGamma_self μ hQ χ.sobolev,hχg,
(Lp.coeFn_nonneg _).mpr (SobolevSpace.minimalSlope_nonneg μ hQ χ.sobolev)] with x hx hg hn
rw [hx]
exact pow_le_pow_left₀ hn hg 2
let F := SobolevSpace.product μ hQ u χ.sobolev A 1
obtain ⟨q,hq,hqr⟩ := hd.product μ hQ u χ.sobolev d χ.laplacian χ.weak_laplacian
A 1 (C/r).toNNReal hA hχb hΓ
have hFr : SobolevSpace.toLp μ hQ F =ᵐ[μ] fun x => χ.weight x*v x := by
filter_upwards [SobolevSpace.product_coe μ hQ u χ.sobolev A 1 hA hχb,hu,χ.representative] with x hx hy hz
change SobolevSpace.toLp μ hQ (SobolevSpace.product μ hQ u χ.sobolev A 1) x = _
rw [hx,hy,hz,mul_comm]
obtain ⟨B,hB⟩ := lipschitz_mul_compact_cutoff χ.lipschitz hv (χ.hasCompactSupport μ hQ)
(fun x => by rw [abs_of_nonneg (χ.bounds x).1]; exact (χ.bounds x).2)
have hFc : F ∈ compactLaplacianCore μ hQ univ :=
⟨⟨_,B,hB,(χ.hasCompactSupport μ hQ).mul_right,subset_univ _,hFr⟩,q,hq⟩
refine ⟨⟨F,hFc⟩,q,hq,?_,?_⟩
· apply (ae_restrict_iff' measurableSet_ball).mpr
filter_upwards [hFr,hu] with x hx hy hball
rw [hx,χ.one x hball,one_mul,hy]
· apply (ae_restrict_iff' measurableSet_ball).mpr
filter_upwards [hqr,χ.representative,χ.laplacian_locality,
localGamma_zero_on_level μ hQ χ.sobolev u 1,
localGamma_comm μ hQ u χ.sobolev] with x hx hχ hΔ hz hc hball
have hχ1 := χ.one x hball
have hΓ0 : localGamma μ hQ u χ.sobolev x=0 := hc.trans (hz (hχ.trans hχ1))
rw [hx,hχ,hχ1,hΔ (Or.inl (by simpa only [mem_ball,dist_comm] using hball)),hΓ0]
ring
end RegularPointCoordinates
end
end OAI
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