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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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Geometry.RCD.Transport.CompactFactorMidpoint
import OAI.Geometry.RCD.Heat.KernelHeatLogarithmicLiYau
import OAI.Geometry.RCD.Heat.HeatLogarithmicSlopeBound
import OAI.Geometry.RCD.Heat.ScaledHamiltonBound
import OAI.Geometry.RCD.Heat.ScaledLiYauSource
namespace OAI
/-! # Uniform lower weak sources for scaled compact-factor logarithms
The source and gradient belong to the same local logarithm. The constants
are independent of the endpoint factor and of its normalization.
-/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped NNReal ENNReal
noncomputable section
namespace RegularPointCoordinates.NoncollapsedRCD
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X)
(hn : 2 ≤ n) (p : X) [SeparableSpace X]
[IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular]
theorem compact_heat_factor_scaled_source (hK : K ≤ 0) :
∃ C D : ℝ, 0 ≤ C ∧ 0 < D ∧
∀ {S : Set X} [CompactSpace S] (ν : Measure S) [IsProbabilityMeasure ν]
(ρ : X → ℝ) (hρ : Measurable ρ) (hρ0 : ∀ x, 0 ≤ ρ x) {B : ℝ≥0},
(∀ᵐ x ∂hausdorff n X, |ρ x| ≤ B) →
ν.map Subtype.val = (hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x)) →
∀ (F : C(S, ℝ)), (∀ a, 0 < F a) →
∀ {ε s δ : ℝ} (hε : 0 < ε) (_hε1 : ε ≤ 1) (hδ : 0 < δ)
(hδs : δ ≤ s) (_hs1 : s ≤ 1),
∀ (z : X) (R : ℝ), 0 ≤ R → (∀ a : S, dist z a ≤ R) →
∀ (Ω : Set X), IsCompact Ω →
∃ (v : SobolevSpace (hausdorff n X) hX.quadratic) (q : Lp ℝ 2 (hausdorff n X)),
IsWeakLaplacian (hausdorff n X) hX.quadratic v q ∧
(SobolevSpace.toLp (hausdorff n X) hX.quadratic v =ᵐ[(hausdorff n X).restrict Ω]
fun x => ε*Real.log (hX.compactHeatFactor hn p ν F (ε*s/2)
(div_pos (mul_pos hε (hδ.trans_le hδs)) (by norm_num)) x)) ∧
∀ᵐ x ∂(hausdorff n X).restrict Ω,
(SobolevSpace.minimalSlope (hausdorff n X) hX.quadratic v x)^2 ≤
Real.exp |K| *(4*C/δ+16*D*(R+dist z x)^2/δ^2) ∧
-(Real.exp |K| *(4*C/δ+16*D*(R+dist z x)^2/δ^2)+
2*(n:ℝ)*|K| +8*(n:ℝ)/δ) ≤ q x := by
obtain ⟨C,D,hC,hD,hdata⟩ := hX.compact_heat_factor_midpoint hn p
refine ⟨C,D,hC,hD,?_⟩
intro S _ ν _ ρ hρ hρ0 B hB hν F hF ε s δ hε hε1 hδ hδs hs1 z R hR hdist Ω hΩ
let m := hausdorff n X
let Q := hX.quadratic
let hk := hX.canonicalHeatKernel_isContinuousHeatKernel hn p
let t := ε*s/4
have hs : 0 < s := hδ.trans_le hδs
have ht : 0 < t := by dsimp only [t]; positivity
have ht1 : t ≤ 1/2 := by
have hm := mul_le_mul hε1 hs1 hs.le (by norm_num : (0:ℝ) ≤ 1)
dsimp only [t]
linarith
have htime : 2*t = ε*s/2 := by dsimp only [t]; ring
obtain ⟨g,A,_hA,hg0,hgA,he,hgap⟩ := hdata ν ρ hρ hρ0 hB hν F hF t ht ht1
have hH (x : X) : kernelHeat m Q hk t g x =
hX.compactHeatFactor hn p ν F (2*t) (by positivity) x := by
simpa only [two_mul] using he t ht x
have hpos (x : X) : 0 < kernelHeat m Q hk t g x := by
rw [hH]
exact hX.compactHeatFactor_positive hn p ν F hF (2*t) (by positivity) x
obtain ⟨u,v,q,hu,hq,hv,hchain,hLY⟩ :=
hX.kernelHeat_logarithmic_liYau hn p hK (1+ε) (by linarith)
hk g hg0 A hgA ht hΩ (fun x _ => hpos x)
have hrep := kernelHeat_represents_L2 m Q hk ht g
have hue : u = heatSobolev m Q t ht g := by
apply Subtype.ext
exact Lp.ext (hu.symm.trans hrep)
have hposAE : ∀ᵐ x ∂m.restrict Ω, 0 < heatAt m Q t g x := by
filter_upwards [ae_restrict_of_ae hrep] with x hx
exact hx ▸ hpos x
have hc (w : SobolevSpace m Q) : localGamma m Q v w =ᵐ[m.restrict Ω]
fun x => localGamma m Q (heatSobolev m Q t ht g) w x/heatAt m Q t g x := by
filter_upwards [hchain w,ae_restrict_of_ae hrep] with x hx hy
rw [hue,hy] at hx
exact hx
have hb := hX.heat_logarithmic_slope_bound hn p g hg0 A hgA ht v hposAE hc
have hgrad : ∀ᵐ x ∂m.restrict Ω, t*Real.exp (-2*|K| *t)*
(SobolevSpace.minimalSlope m Q v x)^2 ≤ C+D*(R+dist z x)^2/t := by
filter_upwards [hb,ae_restrict_of_ae hrep] with x hx hy
have hr := hgap x (R+dist z x) (add_nonneg hR dist_nonneg) (fun a => by
calc
dist x a ≤ dist x z+dist z a := dist_triangle x z a
_ ≤ dist x z+R := add_le_add le_rfl (hdist a)
_ = R+dist z x := by rw [dist_comm x z]; ring)
rw [← hy,hH] at hx
exact hx.trans hr
refine ⟨ε • v,ε • q,hq.smul m Q ε,?_,?_⟩
· filter_upwards [ae_restrict_of_ae (Lp.coeFn_smul ε (SobolevSpace.toLp m Q v)),hv]
with x hx hy
change SobolevSpace.toLp m Q (ε • v) x = ε*SobolevSpace.toLp m Q v x at hx
rw [hx,hy,hH]
simp only [htime]
· have hscale : SobolevSpace.minimalSlope m Q (ε • v) =ᵐ[m]
fun x => ε*SobolevSpace.minimalSlope m Q v x := by
rw [SobolevSpace.minimalSlope_smul_nonneg m Q v hε.le]
exact Lp.coeFn_smul ε (SobolevSpace.minimalSlope m Q v)
filter_upwards [hgrad,hLY,ae_restrict_of_ae hscale,
ae_restrict_of_ae (Lp.coeFn_smul ε q),ae_restrict_of_ae (localGamma_self m Q v)]
with x hx hly hsc hqs hself
have hbound := scaled_hamilton_bound (abs_nonneg K) hC hD.le hε hε1 hδ hδs hs1 rfl hx
rw [hsc]
refine ⟨hbound,?_⟩
change (ε • q) x = ε*q x at hqs
rw [hqs]
have hly' : localGamma m Q v v x ≤ (1+ε)*(q x+localGamma m Q v v x)+
((n:ℝ)*(1+ε)^2/2)*(|K|/ε+1/(ε*s/4)) := by
dsimp only [t] at hly
rw [show 1+ε-1 = ε by ring] at hly
linarith
have hn0 : (0:ℝ) ≤ n := Nat.cast_nonneg n
have hlower := scaled_liYau_source_bound hε hε1 hδ hδs hn0 (abs_nonneg K)
(show 0 ≤ localGamma m Q v v x by rw [hself]; positivity) hly'
rw [hself] at hlower
nlinarith
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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