openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Recursion /CriticalTruncatedPotential.lean
| 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 | |