| import OAI.Analysis.AncientFlat.Geometry.ParabolicWindows |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Laplacian |
| namespace ParabolicLocalization |
| open ParabolicInterior |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| lemma holder_window_smul {α K q : ℝ} (hα : 0≤α) (hK : 0≤K) (hq : 0<q) (hq1 : q≤1) |
| (p0 : V × ℝ) (n : ℕ) {Ω S : Set (V × ℝ)} {f : V × ℝ → E} |
| (hf : HolderBound α Ω f K) (hm : MapsTo (window p0 q) S Ω) : |
| HolderBound α S (fun p => q^n • f (window p0 q p)) K := by |
| apply ((holder_window hα hq hq1 p0 hf hm).const_smul (q^n)).mono |
| rw [abs_of_nonneg (pow_nonneg hq.le n)] |
| exact (mul_le_mul_of_nonneg_right (pow_le_one₀ hq.le hq1) hK).trans_eq (one_mul K) |
| lemma window_jet_bound {α M q : ℝ} (hα : 0≤α) (hM : 0≤M) (hq : 0<q) (hq1 : q≤1) |
| (p0 : V × ℝ) {Ω S : Set (V × ℝ)} (u ut : ℝ → V → ℂ) |
| (hs : ∀ t, ContDiff ℝ 4 (u t)) (hj : JetBound α Ω u ut M) |
| (hm : MapsTo (window p0 q) S Ω) : |
| JetBound α S (windowValue p0 q u) (windowTime p0 q ut) M := by |
| constructor |
| · intro n hn |
| have heq : (fun p : V × ℝ => iteratedFDeriv ℝ n (windowValue p0 q u p.2) p.1)= |
| (fun p => q^n • iteratedFDeriv ℝ n (u (window p0 q p).2) (window p0 q p).1) := |
| funext (window_jet p0 q u hs n hn) |
| rw [heq] |
| exact holder_window_smul hα hM hq hq1 p0 n (hj.1 n hn) hm |
| · exact holder_window_smul hα hM hq hq1 p0 4 hj.2 hm |
| lemma window_top_seminorm {α H q : ℝ} (hq : 0<q) (p0 : V × ℝ) |
| {Ω S : Set (V × ℝ)} (u : ℝ → V → ℂ) (hs : ∀ t, ContDiff ℝ 4 (u t)) |
| (hh : ∀ p∈Ω, ∀ z∈Ω, |
| ‖iteratedFDeriv ℝ 4 (u p.2) p.1-iteratedFDeriv ℝ 4 (u z.2) z.1‖≤H*pdist p z^α) |
| (hm : MapsTo (window p0 q) S Ω) : |
| ∀ p∈S, ∀ z∈S, |
| ‖iteratedFDeriv ℝ 4 (windowValue p0 q u p.2) p.1- |
| iteratedFDeriv ℝ 4 (windowValue p0 q u z.2) z.1‖≤(q^4*q^α*H)*pdist p z^α := by |
| intro p hp z hz |
| rw [window_jet p0 q u hs 4 le_rfl,window_jet p0 q u hs 4 le_rfl,←smul_sub, |
| norm_smul,Real.norm_of_nonneg (pow_nonneg hq.le 4)] |
| have hh := hh _ (hm hp) _ (hm hz) |
| rw [pdist_window p0 hq.le,Real.mul_rpow hq.le (pdist_nonneg p z)] at hh |
| exact (mul_le_mul_of_nonneg_left hh (pow_nonneg hq.le 4)).trans_eq (by ring) |
| end ParabolicLocalization |
|
|
| end |
|
|
| end OAI |
|
|
