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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 38 39 40 41 42 43 44 45 46 47 48 | import OAI.MeasureTheory.Falconer.Recursion.CriticalScalarPotential
import OAI.MeasureTheory.Falconer.Recursion.TruncatedScalarPotential
namespace OAI
/-! # Critical maximum-truncated potentials with logarithmic scale loss -/
namespace Falconer
open Set MeasureTheory
open scoped ENNReal
variable {X : Type*} [MeasurableSpace X]
theorem truncated_scalar_potential_critical (μ : Measure X) (a : X → ℝ)
(ha : Measurable a) (ha0 : ∀ x, 0 ≤ a x)
{K q : ℝ} (hq : 0 ≤ q) (g k : ℕ)
(hsmall : ∀ j : ℕ, j ≤ g+k →
μ {x | a x < 1/(2:ℝ)^j} ≤ ENNReal.ofReal (K*(1/(2:ℝ)^j)^q)) :
(∫⁻ x in {x | a x < 1/(2:ℝ)^g},
ENNReal.ofReal ((1/max (a x) (1/(2:ℝ)^(g+k)))^q) ∂μ) ≤
ENNReal.ofReal ((2:ℝ)^q)*
(ENNReal.ofReal K+(k:ℝ≥0∞)*ENNReal.ofReal (K*(2:ℝ)^q)) := by
have hp (x : X) : (1/max (a x) (1/(2:ℝ)^(g+k)))^q ≤
(2:ℝ)^q*(1/(a x+1/(2:ℝ)^(g+k)))^q := by
have hax := ha0 x
calc
_ ≤ (2/(a x+1/(2:ℝ)^(g+k)))^q := Real.rpow_le_rpow (by positivity)
(inverse_max_le_two_inverse_add (ha0 x) (by positivity)) hq
_ = _ := by
rw [show 2/(a x+1/(2:ℝ)^(g+k)) = 2*(1/(a x+1/(2:ℝ)^(g+k))) by ring,
Real.mul_rpow (by norm_num) (by positivity)]
calc
_ ≤ ∫⁻ x in {x | a x < 1/(2:ℝ)^g},
ENNReal.ofReal ((2:ℝ)^q)*ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ := by
apply lintegral_mono
intro x
dsimp only
rw [← ENNReal.ofReal_mul (by positivity)]
exact ENNReal.ofReal_le_ofReal (hp x)
_ = ENNReal.ofReal ((2:ℝ)^q)*(∫⁻ x in {x | a x < 1/(2:ℝ)^g},
ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ) :=
lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
_ ≤ _ := mul_le_mul' le_rfl
(regularized_scalar_potential_critical μ a ha ha0 hq g k hsmall)
end Falconer
end OAI
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