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12dee50 | 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 | import OAI.Analysis.AncientFlat.Geometry.ExtensionHolder
import OAI.Analysis.AncientFlat.Evolution.FiniteScaling
import OAI.Analysis.AncientFlat.Weighted.ExactXNorm
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff ENNReal
namespace FlatWeighted
open ParabolicInterior
variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V]
lemma holder_normBox_pullback {α w a M : ℝ} {f : V × ℝ → ℂ} {p : V × ℝ}
(hs : sigma p=1) (hh : HolderBound α (normBox a p) f M) :
HolderBound α (localDomain a p) (pullback w p f) M := by
have he (q : V × ℝ) : pullback w p f q=f (chart p q) := by simp only [pullback,hs,Real.one_rpow,one_smul]
constructor
· intro q hq
rw [he]
exact hh.1 _ (mem_image_of_mem _ hq)
· intro q hq r hr
rw [he,he]
have h := hh.2 _ (mem_image_of_mem _ hq) _ (mem_image_of_mem _ hr)
simpa only [chart_pdist,hs,one_mul] using h
lemma extension_YBound {ψ : ℝ → ℝ} (hψ : ExtensionCutoff ψ)
{α w a T M K : ℝ} {f : V × ℝ → ℂ} (hf : FiniteYBound α w a T M f)
(hα : 0≤α) (hw : w≤4) (hT : 0<T) (ha : 0<a) (ha1 : a≤1/16) (hM : 0≤M) (hK : 0≤K)
(hk : ∀ T : ℝ, 0<T → ∀ p : V × ℝ, p.2<0 → sigma p=1 →
HolderBound α (normBox a p) (extensionWeight ψ T) K) :
YBound α w a (2*K*(((1:ℝ)/8)^(w-4)+((1:ℝ)/8)^(w-4)*((1:ℝ)/8)^(-α)+
2*((1:ℝ)/8)^(w-4)/(a*((1:ℝ)/8))^α)*M) (innerExtension ψ T f) := by
intro p hp
obtain ⟨r,hr,q,hq,hs,rfl⟩ := normalize_sigma p hp
have hTr : 0<T/r^4 := div_pos hT (pow_pos hr _)
have hh := (hf.scale r hr).extension_normBox hψ hα hw hTr ha ha1 hM hK q hq hs (hk _ hTr q hq hs)
have ht := holder_normBox_pullback (w := w-4) hs hh
rw [←innerExtension_scale ψ (w-4) T r hr f] at ht
simpa only [←localDomain_scale a r hr q, pullback_scale (w-4) r hr q hq] using ht
theorem innerExtension_bound (ψ : ℝ → ℝ) (hψ : ExtensionCutoff ψ)
(δ α a : ℝ) (hδ : 0<δ) (hδ1 : δ<1) (hα : 0<α) (hα1 : α<1)
(ha : 0<a) (ha1 : a≤1/16) :
∃ C : ℝ, 0<C ∧ ∀ T : ℝ, 0<T → ∀ f : V × ℝ → ℂ,
finiteYNorm α (-2+2*δ) a T f≠⊤ →
yNorm α (-2+2*δ) a (innerExtension ψ T f)≤
ENNReal.ofReal C*finiteYNorm α (-2+2*δ) a T f := by
let w := -2+2*δ
have hw : w ∈ Ioo (-2) 0 := by
dsimp [w]
constructor <;> linarith [hδ, hδ1]
let C0 := ((1:ℝ)/8)^(w-4)+((1:ℝ)/8)^(w-4)*((1:ℝ)/8)^(-α)+2*((1:ℝ)/8)^(w-4)/(a*((1:ℝ)/8))^α
have hC : 0<C0 := by dsimp [C0]; positivity
obtain ⟨K,hK,hk⟩ := extensionWeight_normBox (V := V) hψ α hα hα1.le
refine ⟨4*K*C0,by positivity,?_⟩
intro T hT f hf
have hbound := extension_YBound hψ (finiteYNorm_finite_bound hf) hα.le
(hw.2.le.trans (by norm_num)) hT ha ha1 ENNReal.toReal_nonneg hK.le
(fun T hT p hp hs => hk T a hT ha ha1 p hp hs)
have hN : yNorm α w a (innerExtension ψ T f)≤
ENNReal.ofReal (2*(2*K*C0*(finiteYNorm α w a T f).toReal)) := by
apply iSup_le
intro p
exact holderBound_holderNorm (by positivity) (hbound p.val p.property)
calc
_ ≤ ENNReal.ofReal (2*(2*K*C0*(finiteYNorm α w a T f).toReal)) := hN
_ = ENNReal.ofReal (4*K*C0)*finiteYNorm α w a T f := by
rw [show 2*(2*K*C0*(finiteYNorm α w a T f).toReal)=(4*K*C0)*(finiteYNorm α w a T f).toReal by ring, ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_toReal hf]
end FlatWeighted
end
end OAI
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