| import OAI.Analysis.AncientFlat.Geometry.InteriorSchauder |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Laplacian |
| namespace ParabolicLocalization |
| open ParabolicInterior |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
|
|
| theorem window_center_estimate (α κ K : ℝ) (hα : 0<α) (hα1 : α<1) |
| (hκ : 0<κ) (hK : 0≤K) : |
| ∃ C : ℝ, 0<C ∧ ∀ (q : ℝ), 0<q → q≤1 → |
| ∀ (p0 : V × ℝ) (Ω : Set (V × ℝ)), MapsTo (window p0 q) (cylinder 2) Ω → |
| ∀ (a : V × ℝ → ℝ) (b : V × ℝ → ℂ), |
| HolderBound α Ω a K → HolderBound α Ω b K → (∀ p∈Ω, κ≤a p) → |
| ∀ (u ut f : ℝ → V → ℂ) (F U : ℝ), 0≤F → 0≤U → |
| (∀ t, ContDiff ℝ 4 (u t)) → |
| (∀ p∈Ω, HasDerivAt (fun t => u t p.1) (ut p.2 p.1) p.2) → |
| (∃ M, 0≤M ∧ JetBound α Ω u ut M) → |
| HolderBound α Ω (fun p => f p.2 p.1) F → (∀ p∈Ω, ‖u p.2 p.1‖≤U) → |
| (∀ p∈Ω, ut p.2 p.1+(a p : ℂ)*Δ (Δ (u p.2)) p.1+b p*u p.2 p.1=f p.2 p.1) → |
| (∀ n : ℕ, n≤4 → ‖iteratedFDeriv ℝ n (u p0.2) p0.1‖≤(C/q^4)*(F+U)) ∧ |
| ‖ut p0.2 p0.1‖≤(C/q^4)*(F+U) := by |
| obtain ⟨C,hC,Hlocal⟩ := interior_schauder (V := V) α κ K hα hα1 hκ hK |
| refine ⟨C,hC,?_⟩ |
| intro q hq hq1 p0 Ω hm a b ha hb hell u ut f F U hF hU hs hd hj hf hu he |
| obtain ⟨M,hM,hj⟩ := hj |
| have Hloc := Hlocal (fun p => a (window p0 q p)) (fun p => q^4 • b (window p0 q p)) |
| (holder_window hα.le hq hq1 p0 ha hm) (holder_window_smul hα.le hK hq hq1 p0 4 hb hm) |
| (fun p hp => hell _ (hm hp)) |
| (windowValue p0 q u) (windowTime p0 q ut) (windowTime p0 q f) F U hF hU |
| (windowValue_contDiff p0 q u hs) (fun p hp => window_hasDerivAt p0 q u ut p (hd _ (hm hp))) |
| ⟨M,hM,window_jet_bound hα.le hM hq hq1 p0 u ut hs hj hm⟩ |
| (holder_window_smul hα.le hF hq hq1 p0 4 hf hm) (fun p hp => hu _ (hm hp)) |
| (fun p hp => window_equation p0 q a b u ut f hs p (he _ (hm hp))) |
| have hz : ((0:V),(0:ℝ))∈cylinder 1 := by norm_num [cylinder] |
| have hq4 : 0<q^4 := pow_pos hq _ |
| constructor |
| · intro n hn |
| have hh := (Hloc.1 n hn).1 (0,0) hz |
| dsimp only at hh |
| rw [window_jet p0 q u hs n hn (0,0),window_zero,norm_smul,Real.norm_of_nonneg (pow_nonneg hq.le n)] at hh |
| have hp : q^4≤q^n := pow_le_pow_of_le_one hq.le hq1 hn |
| have hm := (mul_le_mul_of_nonneg_right hp (norm_nonneg (iteratedFDeriv ℝ n (u p0.2) p0.1))).trans hh |
| have hhdiv : ‖iteratedFDeriv ℝ n (u p0.2) p0.1‖≤C*(F+U)/q^4 := |
| (le_div_iff₀ hq4).mpr (by simpa [mul_comm] using hm) |
| convert hhdiv using 1; ring |
| · have hh := Hloc.2.1 (0,0) hz |
| simp only [windowTime,smul_zero,add_zero,mul_zero,norm_smul,Real.norm_of_nonneg hq4.le] at hh |
| have hhdiv : ‖ut p0.2 p0.1‖≤C*(F+U)/q^4 := |
| (le_div_iff₀ hq4).mpr (by simpa [mul_comm] using hh) |
| convert hhdiv using 1; ring |
|
|
| theorem nested_derivative_sup (α κ K : ℝ) (hα : 0<α) (hα1 : α<1) |
| (hκ : 0<κ) (hK : 0≤K) : |
| ∃ C : ℝ, 0<C ∧ ∀ (r R : ℝ), 1≤r → r<R → R≤2 → |
| ∀ (a : V × ℝ → ℝ) (b : V × ℝ → ℂ), |
| HolderBound α (cylinder R) a K → HolderBound α (cylinder R) b K → (∀ p∈cylinder R, κ≤a p) → |
| ∀ (u ut f : ℝ → V → ℂ) (F U : ℝ), 0≤F → 0≤U → |
| (∀ t, ContDiff ℝ 4 (u t)) → |
| (∀ p∈cylinder R, HasDerivAt (fun t => u t p.1) (ut p.2 p.1) p.2) → |
| (∃ M, 0≤M ∧ JetBound α (cylinder R) u ut M) → |
| HolderBound α (cylinder R) (fun p => f p.2 p.1) F → (∀ p∈cylinder R, ‖u p.2 p.1‖≤U) → |
| (∀ p∈cylinder R, ut p.2 p.1+(a p : ℂ)*Δ (Δ (u p.2)) p.1+b p*u p.2 p.1=f p.2 p.1) → |
| ∀ p∈cylinder r, |
| (∀ n : ℕ, n≤4 → ‖iteratedFDeriv ℝ n (u p.2) p.1‖≤(C/(R-r)^4)*(F+U)) ∧ |
| ‖ut p.2 p.1‖≤(C/(R-r)^4)*(F+U) := by |
| obtain ⟨C,hC,Hw⟩ := window_center_estimate (V := V) α κ K hα hα1 hκ hK |
| refine ⟨256*C,by positivity,?_⟩ |
| intro r R hr hrR hR a b ha hb hell u ut f F U hF hU hs hd hj hf hu he p hp |
| let q := (R-r)/4 |
| have hq : 0<q := by dsimp [q]; linarith |
| have hq1 : q≤1 := by dsimp [q]; linarith |
| have hm4 := small_window_maps (by linarith : 0≤r) hq (by dsimp [q]; linarith : 4*q≤R-r) p hp |
| have hm : MapsTo (window p q) (cylinder 2) (cylinder R) := |
| fun z hz => hm4 (cylinder_mono (by norm_num) (by norm_num : (2:ℝ)≤4) hz) |
| have hh := Hw q hq hq1 p (cylinder R) hm a b ha hb hell u ut f F U hF hU hs hd hj hf hu he |
| have heq : C/q^4=256*C/(R-r)^4 := by dsimp [q]; field_simp; ring |
| simpa only [heq] using hh |
| end ParabolicLocalization |
|
|
| end |
|
|
| end OAI |
|
|
