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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 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Angular.CompactContDiffOnDerivativeBound3
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 : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
lemma compact_multiplier_derivative_bound {w : E → ℂ} {K : Set E}
(hK : IsCompact K) (hw : ContDiff ℝ ∞ w) (m : ℕ) (C : ℝ) (hC : 0 ≤ C) :
∃ D : ℝ, 0 ≤ D ∧ ∀ (b : E → ℂ) (A : ℝ), ContDiff ℝ ∞ b → 1 ≤ A →
(∀ j ≤ m, ∀ x ∈ K, ‖iteratedFDeriv ℝ j b x‖ ≤ C*A^j) →
∀ j ≤ m, ∀ x ∈ K, ‖iteratedFDeriv ℝ j (fun x => w x*b x) x‖ ≤ D*A^j := by
obtain ⟨W,hW,hWw⟩ := compact_contDiffOn_derivative_bound isOpen_univ hK
(subset_univ K) hw.contDiffOn m
have hW0 : 0 ≤ W := zero_le_one.trans hW
refine ⟨2^m*W*C,by positivity,?_⟩
intro b A hb hA hderiv j hj x hx
have hA0 := zero_le_one.trans hA
have hi : (j : WithTop ℕ∞) ≤ (↑(⊤:ℕ∞) : WithTop ℕ∞) := by
exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤)
apply (norm_iteratedFDeriv_mul_le hw hb x hi).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)*‖iteratedFDeriv ℝ i w x‖*
‖iteratedFDeriv ℝ (j-i) b x‖) ≤
∑ i ∈ Finset.range (j+1), (j.choose i:ℝ)*W*(C*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 hw' : ‖iteratedFDeriv ℝ i w x‖ ≤ W := by
simpa only [iteratedFDerivWithin_univ] using hWw i (hij.trans hj) x hx
apply mul_le_mul
(mul_le_mul_of_nonneg_left hw' (by positivity))
((hderiv (j-i) ((Nat.sub_le _ _).trans hj) x hx).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hA (Nat.sub_le _ _)) hC))
(norm_nonneg _) (by positivity)
_ = 2^j*W*(C*A^j) := by rw [← Finset.sum_mul,← Finset.sum_mul,hsum]
_ ≤ 2^m*W*(C*A^j) := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) hj) hW0)
(by positivity)
_ = _ := by ring
end Attempt4PrefixShell
end GraphLiteralSupport
end
end
end OAI
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