| import OAI.Analysis.AncientFlat.Geometry.WindowJets |
| import OAI.Analysis.AncientFlat.Regularity.HolderProducts |
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|
| 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] |
| lemma holder_window_general {α K q : ℝ} (hK : 0≤K) (hq : 0<q) (p0 : V × ℝ) |
| {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 => f (window p0 q p)) ((1+q^α)*K) := by |
| have hqα : 0≤q^α := Real.rpow_nonneg hq.le _ |
| refine ⟨fun p hp => (hf.1 _ (hm hp)).trans (by nlinarith),?_⟩ |
| intro p hp z hz |
| have hh := hf.2 _ (hm hp) _ (hm hz) |
| rw [pdist_window p0 hq.le,Real.mul_rpow hq.le (pdist_nonneg p z)] at hh |
| nlinarith [mul_nonneg hK (Real.rpow_nonneg (pdist_nonneg p z) α)] |
| lemma power_le_one_add_four {q : ℝ} (hq : 0≤q) (n : ℕ) (hn : n≤4) : q^n≤1+q^4 := by |
| by_cases hq1 : q≤1 |
| · exact (pow_le_one₀ hq hq1).trans (le_add_of_nonneg_right (pow_nonneg hq 4)) |
| · exact (pow_le_pow_right₀ (not_le.mp hq1).le hn).trans (by linarith) |
| lemma window_jet_bound_general {α M q : ℝ} (hM : 0≤M) (hq : 0<q) |
| (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) ((1+q^4)*(1+q^α)*M) := by |
| have H {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (n : ℕ) (hn : n≤4) (f : V × ℝ → E) (hf : HolderBound α Ω f M) : |
| HolderBound α S (fun p => q^n • f (window p0 q p)) ((1+q^4)*(1+q^α)*M) := by |
| apply ((holder_window_general hM hq p0 hf hm).const_smul (q^n)).mono |
| rw [abs_of_nonneg (pow_nonneg hq.le _)] |
| nlinarith [mul_le_mul_of_nonneg_right (power_le_one_add_four hq.le n hn) |
| (show 0≤(1+q^α)*M by positivity)] |
| 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 H n hn _ (hj.1 n hn) |
| · exact H 4 le_rfl _ hj.2 |
| end ParabolicLocalization |
| namespace ParabolicInterior |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| lemma JetBound.const_smul {α M : ℝ} {Ω : Set (V × ℝ)} {u ut : ℝ → V → ℂ} |
| (h : JetBound α Ω u ut M) (hs : ∀ t, ContDiff ℝ 4 (u t)) (c : ℝ) : |
| JetBound α Ω (fun t x => c • u t x) (fun t x => c • ut t x) (|c| * M) := by |
| constructor |
| · intro n hn |
| have he : (fun p : V × ℝ => iteratedFDeriv ℝ n (fun x => c • u p.2 x) p.1)= |
| (fun p => c • iteratedFDeriv ℝ n (u p.2) p.1) := by |
| funext p |
| exact iteratedFDeriv_const_smul_apply' ((hs p.2).of_le (by exact_mod_cast hn)).contDiffAt |
| rw [he] |
| exact (h.1 n hn).const_smul c |
| · exact h.2.const_smul c |
| end ParabolicInterior |
|
|
| end |
|
|
| end OAI |
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