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b224933 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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