openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Shells /IntegralDyadicShellBound.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.LintegralBiUnionFinsetLe | |
| namespace OAI | |
| /-! Dyadic shells and prefix decompositions. -/ | |
| section | |
| section | |
| namespace CriticalFourierSupport | |
| namespace CurrentFalconer | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| theorem integral_dyadic_shell_bound (f : ℝ → ℝ) (hf : Continuous f) | |
| (hnonneg : ∀ s, 0 ≤ f s) {H A : ℝ} (hH : 1 ≤ H) (hA : 0 ≤ A) | |
| (hlow : (∫ s : ℝ in Set.Icc (-1) 1, f s) ≤ 2) | |
| (hshell : ∀ B : ℝ, 1 ≤ B → B ≤ H → | |
| (∫ s : ℝ in annularScalarSet B H, f s) ≤ A) : | |
| (∫ s : ℝ in Set.Icc (-H) H, f s) ≤ | |
| 2 + (Real.log H / Real.log 2 + 1) * A := by | |
| obtain ⟨n, hn, hHn⟩ := exists_nat_pow_near hH (by norm_num : (1 : ℝ) < 2) | |
| have hi : IntegrableOn f (Set.Icc (-H) H) := hf.integrableOn_Icc | |
| have hi0 : IntegrableOn f (Set.Icc (-1) 1) := hf.integrableOn_Icc | |
| have hishell (B : ℝ) : IntegrableOn f (annularScalarSet B H) := by | |
| apply hi.mono_set | |
| intro s hs | |
| exact abs_le.mp (hs.2.trans (min_le_right _ _)) | |
| have hsh (j : ℕ) (hj : j ∈ Finset.range (n+1)) : | |
| (∫⁻ s : ℝ in annularScalarSet ((2 : ℝ)^j) H, ENNReal.ofReal (f s)) ≤ | |
| ENNReal.ofReal A := by | |
| rw [← ofReal_integral_eq_lintegral_ofReal (hishell _) (ae_of_all _ hnonneg)] | |
| apply ENNReal.ofReal_le_ofReal (hshell _ (one_le_pow₀ (by norm_num)) _) | |
| exact (pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) | |
| (Nat.le_of_lt_succ (Finset.mem_range.mp hj))).trans hn | |
| have hlow' : (∫⁻ s : ℝ in Set.Icc (-1) 1, ENNReal.ofReal (f s)) ≤ | |
| ENNReal.ofReal 2 := by | |
| rw [← ofReal_integral_eq_lintegral_ofReal hi0 (ae_of_all _ hnonneg)] | |
| exact ENNReal.ofReal_le_ofReal hlow | |
| have hsum : (∑ _j ∈ Finset.range (n+1), ENNReal.ofReal A) = | |
| ENNReal.ofReal (((n : ℝ)+1)*A) := by | |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun _ _ => hA)] | |
| simp | |
| have hbound : ENNReal.ofReal (∫ s : ℝ in Set.Icc (-H) H, f s) ≤ | |
| ENNReal.ofReal (2+((n : ℝ)+1)*A) := by | |
| rw [ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ hnonneg)] | |
| calc | |
| _ ≤ ∫⁻ s : ℝ in Set.Icc (-1) 1 ∪ | |
| ⋃ j ∈ Finset.range (n+1), annularScalarSet ((2 : ℝ)^j) H, | |
| ENNReal.ofReal (f s) := lintegral_mono_set (dyadic_scalar_cover hHn) | |
| _ ≤ (∫⁻ s : ℝ in Set.Icc (-1) 1, ENNReal.ofReal (f s)) + | |
| ∫⁻ s : ℝ in ⋃ j ∈ Finset.range (n+1), annularScalarSet ((2 : ℝ)^j) H, | |
| ENNReal.ofReal (f s) := lintegral_union_le _ _ _ | |
| _ ≤ ENNReal.ofReal 2 + ∑ j ∈ Finset.range (n+1), | |
| ∫⁻ s : ℝ in annularScalarSet ((2 : ℝ)^j) H, ENNReal.ofReal (f s) := | |
| add_le_add hlow' (lintegral_biUnion_finset_le _ _ _ _) | |
| _ ≤ ENNReal.ofReal 2 + ∑ _j ∈ Finset.range (n+1), ENNReal.ofReal A := | |
| add_le_add le_rfl (Finset.sum_le_sum hsh) | |
| _ = _ := by | |
| rw [hsum, ← ENNReal.ofReal_add (by norm_num) (by positivity)] | |
| have hlog2 : 0 < Real.log 2 := Real.log_pos (by norm_num) | |
| have hnlog : (n : ℝ) ≤ Real.log H / Real.log 2 := by | |
| apply (le_div_iff₀ hlog2).mpr | |
| have h := Real.log_le_log (by positivity) hn | |
| simpa only [Real.log_pow] using h | |
| apply ((ENNReal.ofReal_le_ofReal_iff (by positivity)).mp hbound).trans | |
| exact add_le_add le_rfl (mul_le_mul_of_nonneg_right (by linarith : | |
| (n : ℝ)+1 ≤ Real.log H / Real.log 2 + 1) hA) | |
| end CurrentFalconer | |
| end CriticalFourierSupport | |
| end | |
| end | |
| end OAI | |