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b37e936 | 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 | import OAI.Geometry.RCD.Heat.LocalLogarithmicLaplacian
namespace OAI
/-! # Bounded Lipschitz logarithmic extensions with their actual source -/
open Set Filter MeasureTheory
open scoped ENNReal NNReal
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] (μ : Measure X) [IsFiniteMeasureOnCompacts μ] (Q : QuadraticCheegerEnergy μ)
theorem bounded_local_logarithm (u : SobolevSpace μ Q) (d : Lp ℝ 2 μ)
(hd : IsWeakLaplacian μ Q u d) (F : X → ℝ) {L A : ℝ≥0}
(hF : LipschitzWith L F) (hA : ∀ x, |F x|≤A)
(hu : SobolevSpace.toLp μ Q u =ᵐ[μ] F)
{Ω : Set X} {a b : ℝ} (ha : 0<a) (hab : a≤b)
(hv : ∀ᵐ x ∂μ.restrict Ω, F x∈Icc a b) :
∃ (v : SobolevSpace μ Q) (q : Lp ℝ 2 μ) (V : X → ℝ) (M B : ℝ≥0),
IsWeakLaplacian μ Q v q ∧ LipschitzWith M V ∧
(∀ x, |V x|≤B) ∧ (SobolevSpace.toLp μ Q v =ᵐ[μ] V) ∧
(SobolevSpace.toLp μ Q v =ᵐ[μ.restrict Ω] fun x => Real.log (F x)) := by
obtain ⟨φ,ψ,χ,C,D,B,E,hφ,hψ,hχ,hφ0,hψ0,hdφ,hdψ,hB,hE,he⟩ :=
exists_local_smooth_log ha hab
let U := SobolevSpace.toLp μ Q u
have hp : MemLp (fun x => ψ (U x)) ∞ μ := memLp_top_of_bound
(hψ.continuous.comp_aestronglyMeasurable (Lp.aestronglyMeasurable U)) B
(Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hB (U x))
have hc : MemLp (fun x => χ (U x)) ∞ μ := memLp_top_of_bound
(hχ.comp_aestronglyMeasurable (Lp.aestronglyMeasurable U)) E
(Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hE (U x))
have hL := SobolevSpace.minimalSlope_le_lipschitz μ Q u hF hu
have hq : MemLp (fun x => ψ (U x)*d x+χ (U x)*localGamma μ Q u u x) 2 μ :=
(hp.mul (Lp.memLp d)).add (hc.mul (localGamma_memLp_of_bounded_gradient μ Q u u L hL))
let q := hq.toLp _
let v := SobolevSpace.comp μ Q hφ hφ0 u
have hrep : SobolevSpace.toLp μ Q v =ᵐ[μ] fun x => φ (F x) := by
filter_upwards [SobolevSpace.comp_coe μ Q hφ hφ0 u,hu] with x hx hy
exact hx.trans (congrArg φ hy)
refine ⟨v,q,φ ∘ F,C*L,C*A,
weakLaplacian_comp_C2 μ Q u d q hd hφ hφ0 hψ hψ0 hχ hdφ hdψ B E hB hE hq.coeFn_toLp,
hφ.comp hF,?_,hrep,?_⟩
· intro x
have hh := hφ.dist_le_mul (F x) 0
simp only [hφ0,Real.dist_eq,sub_zero] at hh
exact hh.trans (by simpa only [NNReal.coe_mul] using
mul_le_mul_of_nonneg_left (hA x) C.coe_nonneg)
· filter_upwards [ae_restrict_of_ae hrep,hv] with x hx hy
exact hx.trans (he (F x) hy).1
end RegularPointCoordinates
end
end OAI
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