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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 | import OAI.Analysis.AncientFlat.Geometry.WindowJets
import OAI.Analysis.AncientFlat.Regularity.HolderProducts
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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