File size: 1,499 Bytes
d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Angular.StationaryChartAmplitude
namespace OAI
namespace PlanarFalconer
open Set
open scoped ContDiff
theorem compact_multiplier_uniform_derivatives {w : ℝ → ℂ}
(hw : ContDiff ℝ ∞ w) {K : Set ℝ} (hK : IsCompact K) (hs : tsupport w ⊆ K) (n : ℕ) :
∃ A : ℝ, 0 ≤ A ∧ ∀ {f : ℝ → ℂ}, ContDiff ℝ ∞ f →
∀ {C D : ℝ}, 0 ≤ C → 1 ≤ D →
(∀ k ≤ n, ∀ x, ‖iteratedDeriv k f x‖ ≤ C*D^k) →
∀ k ≤ n, ∀ x, ‖iteratedDeriv k (fun y => w y*f y) x‖ ≤ (A*C)*D^k := by
choose A hA hbound using fun k => compact_weighted_composition_scale hw
(contDiff_id : ContDiff ℝ ∞ (id : ℝ → ℝ)) hK k
let B : ℝ := ∑ k ∈ Finset.range (n+1), A k
refine ⟨B,Finset.sum_nonneg (fun k _ => hA k),?_⟩
intro f hf C D hC hD hb k hk x
have hkB : A k ≤ B := Finset.single_le_sum (fun l _ => hA l)
(Finset.mem_range.mpr (Nat.lt_succ_of_le hk))
by_cases hx : x ∈ K
· have hh := hbound k f hf C D hC hD (fun l hl => hb l (hl.trans hk)) x hx
change ‖iteratedDeriv k (fun y => w y*f y) x‖ ≤ A k*C*D^k at hh
apply hh.trans
gcongr
· have he : iteratedDeriv k (fun y => w y*f y) x = 0 :=
image_eq_zero_of_notMem_tsupport (fun hh => hx (hs (tsupport_mul_subset_left
(complex_iteratedDeriv_tsupport _ k hh))))
rw [he,norm_zero]
have hB : 0 ≤ B := Finset.sum_nonneg (fun k _ => hA k)
positivity
end PlanarFalconer
end OAI
|