openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /CompactMultiplierDerivativeShift.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereAmbient2 | |
| 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_shiftND {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) (s : ℕ), ContDiff ℝ ∞ b → 1 ≤ A → | |
| (∀ j ≤ m, ∀ x ∈ K, ‖iteratedFDeriv ℝ j b x‖ ≤ C*A^(j+s)) → | |
| ∀ j ≤ m, ∀ x ∈ K, ‖iteratedFDeriv ℝ j (fun x => w x*b x) x‖ ≤ D*A^(j+s) := 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 s 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+s)) := 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.add_le_add_right (Nat.sub_le _ _) s)) hC)) | |
| (norm_nonneg _) (by positivity) | |
| _ = 2^j*W*(C*A^(j+s)) := by rw [← Finset.sum_mul,← Finset.sum_mul,hsum] | |
| _ ≤ 2^m*W*(C*A^(j+s)) := 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 | |