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7f37411 | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.NearGrid
namespace OAI
/-! Angular cutoffs and derivative estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4PrefixShell
open Set Metric
open scoped ContDiff Topology
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
lemma compact_contDiffOn_derivative_bound {f : E → F} {U K : Set E}
(hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) (hf : ContDiffOn ℝ ∞ f U)
(m : ℕ) : ∃ C : ℝ, 1 ≤ C ∧ ∀ j ≤ m, ∀ x ∈ K,
‖iteratedFDerivWithin ℝ j f U x‖ ≤ C := by
have hbound (j : ℕ) : ∃ C : ℝ, ∀ x ∈ K,
‖iteratedFDerivWithin ℝ j f U x‖ ≤ C :=
hK.exists_bound_of_continuousOn
((hf.continuousOn_iteratedFDerivWithin (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤)) hU.uniqueDiffOn).mono hKU)
induction m with
| zero =>
obtain ⟨C,hC⟩ := hbound 0
refine ⟨max 1 C, le_max_left _ _, ?_⟩
intro j hj x hx
have : j=0 := Nat.eq_zero_of_le_zero hj
subst j
exact (hC x hx).trans (le_max_right _ _)
| succ m ih =>
obtain ⟨D,hD,hboundD⟩ := ih
obtain ⟨C,hC⟩ := hbound (m+1)
refine ⟨max D C, hD.trans (le_max_left _ _), ?_⟩
intro j hj x hx
by_cases hjm : j ≤ m
· exact (hboundD j hjm x hx).trans (le_max_left _ _)
· have : j=m+1 := by omega
subst j
exact (hC x hx).trans (le_max_right _ _)
theorem compact_chart_derivative_bound
{f : E → F} {w : E → ℂ} {U K : Set E}
(hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U)
(hf : ContDiffOn ℝ ∞ f U) (hw : ContDiffOn ℝ ∞ w U)
(m : ℕ) (C : ℝ) (hC : 0 ≤ C) :
∃ D : ℝ, 0 ≤ D ∧ ∀ (b : F → ℂ) (A : ℝ),
ContDiff ℝ ∞ b → 1 ≤ A →
tsupport (fun x => w x * b (f x)) ⊆ K →
(∀ j ≤ m, ∀ y ∈ f '' K, ‖iteratedFDeriv ℝ j b y‖ ≤ C*A^j) →
∀ j ≤ m, ∀ x, ‖iteratedFDeriv ℝ j (fun x => w x * b (f x)) x‖ ≤ D*A^j := by
obtain ⟨M,hM,hMf⟩ := compact_contDiffOn_derivative_bound hU hK hKU hf m
obtain ⟨W,hW,hWw⟩ := compact_contDiffOn_derivative_bound hU hK hKU hw m
let B : ℝ := (m.factorial:ℝ)*C*M^m
let D : ℝ := 2^m * W * B
have hM0 : 0 ≤ M := le_trans zero_le_one hM
have hW0 : 0 ≤ W := le_trans zero_le_one hW
have hB : 0 ≤ B := by dsimp [B]; positivity
refine ⟨D, by dsimp [D]; positivity, ?_⟩
intro b A hb hA hsupp hderiv j hj x
have hA0 : 0 ≤ A := le_trans zero_le_one hA
by_cases hx : x ∈ K
· have hxU := hKU hx
have hcomp (k : ℕ) (hkm : k ≤ m) :
‖iteratedFDerivWithin ℝ k (b ∘ f) U x‖ ≤ B*A^k := by
have hbase := norm_iteratedFDerivWithin_comp_le (N := ∞) (t := univ)
hb.contDiffOn hf (by exact_mod_cast (le_top : (k:ℕ∞) ≤ ⊤)) uniqueDiffOn_univ hU.uniqueDiffOn
(fun _ _ => mem_univ _) hxU (C := C*A^k) (D := M)
(fun i hik => by
rw [iteratedFDerivWithin_univ]
exact (hderiv i (hik.trans hkm) (f x) ⟨x,hx,rfl⟩).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hA hik) hC))
(fun i hi hik => (hMf i (hik.trans hkm) x hx).trans
(by simpa using pow_le_pow_right₀ hM hi))
apply hbase.trans
calc
(k.factorial:ℝ)*(C*A^k)*M^k ≤ (m.factorial:ℝ)*(C*A^k)*M^m := by
gcongr
_ = B*A^k := by dsimp [B]; ring
have hbg : ContDiffOn ℝ ∞ (b ∘ f) U := hb.comp_contDiffOn hf
have hprod : ContDiffOn ℝ ∞ (fun x => w x * b (f x)) U := hw.mul hbg
rw [← iteratedFDerivWithin_eq_iteratedFDeriv hU.uniqueDiffOn
((hprod.contDiffAt (hU.mem_nhds hxU)).of_le (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤))) hxU]
have hbase := norm_iteratedFDerivWithin_mul_le hw hbg hU.uniqueDiffOn hxU
(by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤))
apply hbase.trans
have hsum : (∑ i ∈ Finset.range (j+1), (j.choose i:ℝ)) = 2^j := by
exact_mod_cast Nat.sum_range_choose j
calc
(∑ i ∈ Finset.range (j+1), (j.choose i:ℝ) * ‖iteratedFDerivWithin ℝ i w U x‖ *
‖iteratedFDerivWithin ℝ (j-i) (b ∘ f) U x‖) ≤
∑ i ∈ Finset.range (j+1), (j.choose i:ℝ) * W * (B*A^j) := by
apply Finset.sum_le_sum
intro i hi
have hij : i ≤ j := Nat.le_of_lt_succ (Finset.mem_range.mp hi)
have hk : j-i ≤ m := (Nat.sub_le _ _).trans hj
apply mul_le_mul
(mul_le_mul_of_nonneg_left (hWw i (hij.trans hj) x hx) (by positivity))
((hcomp (j-i) hk).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hA (Nat.sub_le _ _)) hB))
(norm_nonneg _) (by positivity)
_ = 2^j * W * (B*A^j) := by rw [← Finset.sum_mul, ← Finset.sum_mul, hsum]
_ ≤ 2^m * W * (B*A^j) :=
mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
(pow_le_pow_right₀ (show (1:ℝ) ≤ 2 by norm_num) hj) hW0)
(mul_nonneg hB (pow_nonneg hA0 _))
_ = D*A^j := by dsimp [D]; ring
· have hx' : x ∉ tsupport (fun x => w x * b (f x)) := fun h => hx (hsupp h)
have hz : iteratedFDeriv ℝ j (fun x => w x * b (f x)) x = 0 :=
Function.notMem_support.mp (fun h => hx' (support_iteratedFDeriv_subset j h))
rw [hz, norm_zero]
dsimp [D]
positivity
end Attempt4PrefixShell
end GraphLiteralSupport
end
end
end OAI
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