| 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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|
