| import OAI.MeasureTheory.Falconer.Angular.StationaryChartAmplitude |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set |
| open scoped ContDiff |
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|
| 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 |
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| end PlanarFalconer |
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| end OAI |
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