| import OAI.Analysis.AncientFlat.Regularity.CoreHolderJets |
| import OAI.Analysis.AncientFlat.Geometry.InteriorCore |
| import OAI.Analysis.AncientFlat.Regularity.LowerOrderHolder |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Laplacian |
| namespace CompactCoreFinite |
| open CompactCore |
| open ParabolicInterior |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| omit [FiniteDimensional ℝ V] in |
| lemma apply_jet_product_bound (f : V → E →L[ℝ] F) (g : V → E) |
| (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) |
| (x : V) (A B : ℝ) (hA : 0≤A) (hB : 0≤B) |
| (hb : ∀ j : ℕ, j≤2 → ‖iteratedFDeriv ℝ j f x‖≤A) |
| (hc : ∀ j : ℕ, j≤2 → ‖iteratedFDeriv ℝ j g x‖≤B) : |
| ∀ j : ℕ, j≤2 → ‖iteratedFDeriv ℝ j (fun y => f y (g y)) x‖≤4*A*B := by |
| intro j hj |
| have he := norm_iteratedFDeriv_clm_apply hf hg x |
| (by exact_mod_cast hj) |
| apply he.trans |
| calc |
| _ ≤ ∑ i∈Finset.range (j+1), ((j.choose i:ℕ):ℝ)*A*B := by |
| apply Finset.sum_le_sum |
| intro i hi |
| have hi' : i≤j := Nat.le_of_lt_succ (Finset.mem_range.mp hi) |
| exact mul_le_mul |
| (mul_le_mul_of_nonneg_left (hb i (by omega)) (Nat.cast_nonneg (j.choose i))) |
| (hc (j-i) (by omega)) (norm_nonneg _) (by positivity) |
| _ ≤ 4*A*B := by |
| interval_cases j <;> norm_num [Finset.sum_range_succ,Nat.choose] <;> |
| nlinarith [mul_nonneg hA hB] |
| omit [FiniteDimensional ℝ V] in |
| lemma apply_jet_holder {α A B : ℝ} {Ω : Set (V × ℝ)} |
| (f : ℝ → V → E →L[ℝ] F) (g : ℝ → V → E) |
| (hf : ∀ p∈Ω, ContDiff ℝ 2 (f p.2)) (hg : ∀ p∈Ω, ContDiff ℝ 2 (g p.2)) |
| (hA : 0≤A) (hB : 0≤B) |
| (hb : ∀ j : ℕ, j≤2 → HolderBound α Ω (fun p => iteratedFDeriv ℝ j (f p.2) p.1) A) |
| (hc : ∀ j : ℕ, j≤2 → HolderBound α Ω (fun p => iteratedFDeriv ℝ j (g p.2) p.1) B) : |
| ∀ j : ℕ, j≤2 → HolderBound α Ω |
| (fun p => iteratedFDeriv ℝ j (fun y => f p.2 y (g p.2 y)) p.1) (8*A*B) := by |
| intro j hj |
| constructor |
| · intro p hp |
| exact (apply_jet_product_bound _ _ (hf p hp) (hg p hp) p.1 A B hA hB |
| (fun i hi => (hb i hi).1 p hp) (fun i hi => (hc i hi).1 p hp) j hj).trans (by nlinarith [mul_nonneg hA hB]) |
| · intro p hp q hq |
| let f₁ : V → E →L[ℝ] F := fun z => f p.2 (z+p.1) |
| let f₂ : V → E →L[ℝ] F := fun z => f q.2 (z+q.1) |
| let g₁ : V → E := fun z => g p.2 (z+p.1) |
| let g₂ : V → E := fun z => g q.2 (z+q.1) |
| have hf₁ : ContDiff ℝ 2 f₁ := (hf p hp).comp (contDiff_id.add contDiff_const) |
| have hf₂ : ContDiff ℝ 2 f₂ := (hf q hq).comp (contDiff_id.add contDiff_const) |
| have hg₁ : ContDiff ℝ 2 g₁ := (hg p hp).comp (contDiff_id.add contDiff_const) |
| have hg₂ : ContDiff ℝ 2 g₂ := (hg q hq).comp (contDiff_id.add contDiff_const) |
| let d := pdist p q^α |
| have hd : 0≤d := Real.rpow_nonneg (by unfold pdist; positivity) _ |
| have hs : (fun z => f₁ z (g₁ z))-(fun z => f₂ z (g₂ z))= |
| (fun z => (f₁-f₂) z (g₁ z))+(fun z => f₂ z ((g₁-g₂) z)) := by |
| funext z |
| simp only [Pi.sub_apply,Pi.add_apply,sub_apply,map_sub] |
| abel |
| have h₁ : ‖iteratedFDeriv ℝ j (fun z => (f₁-f₂) z (g₁ z)) 0‖≤4*(A*d)*B := by |
| apply apply_jet_product_bound _ _ (hf₁.sub hf₂) hg₁ 0 (A*d) B (mul_nonneg hA hd) hB _ _ j hj |
| · intro i hi |
| change ‖iteratedFDeriv ℝ i (f₁-f₂) 0‖ ≤ A*d |
| rw [iteratedFDeriv_sub (hf₁.of_le (by exact_mod_cast hi)) |
| (hf₂.of_le (by exact_mod_cast hi)),Pi.sub_apply] |
| simpa only [f₁,f₂,iteratedFDeriv_comp_add_right,zero_add] using (hb i hi).2 p hp q hq |
| · intro i hi |
| simpa only [g₁,iteratedFDeriv_comp_add_right,zero_add] using (hc i hi).1 p hp |
| have h₂ : ‖iteratedFDeriv ℝ j (fun z => f₂ z ((g₁-g₂) z)) 0‖≤4*A*(B*d) := by |
| apply apply_jet_product_bound _ _ hf₂ (hg₁.sub hg₂) 0 A (B*d) hA (mul_nonneg hB hd) _ _ j hj |
| · intro i hi |
| simpa only [f₂,iteratedFDeriv_comp_add_right,zero_add] using (hb i hi).1 q hq |
| · intro i hi |
| change ‖iteratedFDeriv ℝ i (g₁-g₂) 0‖ ≤ B*d |
| rw [iteratedFDeriv_sub (hg₁.of_le (by exact_mod_cast hi)) |
| (hg₂.of_le (by exact_mod_cast hi)),Pi.sub_apply] |
| simpa only [g₁,g₂,iteratedFDeriv_comp_add_right,zero_add] using (hc i hi).2 p hp q hq |
| have hsub := iteratedFDeriv_sub (i := j) |
| ((hf₁.clm_apply hg₁).of_le (by exact_mod_cast hj)) |
| ((hf₂.clm_apply hg₂).of_le (by exact_mod_cast hj)) |
| have hadd := iteratedFDeriv_add (i := j) |
| (((hf₁.sub hf₂).clm_apply hg₁).of_le (by exact_mod_cast hj)) |
| ((hf₂.clm_apply (hg₁.sub hg₂)).of_le (by exact_mod_cast hj)) |
| have hh : iteratedFDeriv ℝ j (fun y => f p.2 y (g p.2 y)) p.1- |
| iteratedFDeriv ℝ j (fun y => f q.2 y (g q.2 y)) q.1= |
| iteratedFDeriv ℝ j (fun z => (f₁-f₂) z (g₁ z)) 0+ |
| iteratedFDeriv ℝ j (fun z => f₂ z ((g₁-g₂) z)) 0 := by |
| have he := congrFun hsub 0 |
| rw [hs] at he |
| simp only [Pi.sub_apply] at he |
| rw [hadd,Pi.add_apply] at he |
| have he₁ := iteratedFDeriv_comp_add_right (𝕜 := ℝ) (f := fun y => f p.2 y (g p.2 y)) j p.1 0 |
| have he₂ := iteratedFDeriv_comp_add_right (𝕜 := ℝ) (f := fun y => f q.2 y (g q.2 y)) j q.1 0 |
| change iteratedFDeriv ℝ j (fun x => f p.2 (x+p.1) (g p.2 (x+p.1))) 0 = _ at he₁ |
| change iteratedFDeriv ℝ j (fun x => f q.2 (x+q.1) (g q.2 (x+q.1))) 0 = _ at he₂ |
| dsimp only [f₁,f₂,g₁,g₂] at he ⊢ |
| simp only [zero_add] at he₁ he₂ |
| rw [he₁,he₂] at he |
| exact he.symm |
| rw [hh] |
| exact (norm_add_le _ _).trans ((add_le_add h₁ h₂).trans_eq (by dsimp [d]; ring)) |
| end CompactCoreFinite |
|
|
| end |
|
|
| end OAI |
|
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