| import OAI.MeasureTheory.Falconer.Recursion.PotentialFrostman |
|
|
| namespace OAI |
|
|
| /-! # The logarithmic potential bound at the critical ball exponent |
|
|
| Unlike the strictly subcritical energy estimate, the finite critical estimate |
| has one bounded contribution per dyadic scale. It is used on the outer |
| intervals of the radial profile argument. |
| -/ |
|
|
| namespace Falconer |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
| variable {X : Type*} [PseudoMetricSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] |
|
|
| theorem regularized_potential_critical (μ : Measure X) (x : X) |
| {K q : ℝ} (hq : 0 ≤ q) (g k : ℕ) |
| (hball : ∀ n : ℕ, n ≤ g+k → |
| μ (ball x (1/(2:ℝ)^n)) ≤ ENNReal.ofReal (K*(1/(2:ℝ)^n)^q)) : |
| (∫⁻ y in ball x (1/(2:ℝ)^g), |
| ENNReal.ofReal ((1/(dist x y+1/(2:ℝ)^(g+k)))^q) ∂μ) ≤ |
| ENNReal.ofReal K+(k:ℝ≥0∞)*ENNReal.ofReal (K*(2:ℝ)^q) := by |
| apply (regularized_potential_dyadic μ x hq g k).trans |
| have hstart : ENNReal.ofReal (((2:ℝ)^g)^q)*μ (ball x (1/(2:ℝ)^g)) ≤ |
| ENNReal.ofReal K := by |
| calc |
| _ ≤ ENNReal.ofReal (((2:ℝ)^g)^q)*ENNReal.ofReal (K*(1/(2:ℝ)^g)^q) := |
| mul_le_mul_right (hball g (by omega)) _ |
| _ = _ := by |
| rw [← ENNReal.ofReal_mul (by positivity)] |
| congr 1 |
| calc |
| _ = K*((((2:ℝ)^g)^q)*(1/(2:ℝ)^g)^q) := by ring |
| _ = K := by rw [dyadic_potential_start,sub_self,Real.rpow_zero,mul_one] |
| have hterm (j : ℕ) (hj : j ∈ Finset.range k) : |
| ENNReal.ofReal (((2:ℝ)^(g+j+1))^q)*μ (ball x (1/(2:ℝ)^(g+j))) ≤ |
| ENNReal.ofReal (K*(2:ℝ)^q) := by |
| calc |
| _ ≤ ENNReal.ofReal (((2:ℝ)^(g+j+1))^q)* |
| ENNReal.ofReal (K*(1/(2:ℝ)^(g+j))^q) := |
| mul_le_mul_right (hball (g+j) (by have := Finset.mem_range.mp hj |
| _ = _ := by |
| rw [← ENNReal.ofReal_mul (by positivity)] |
| congr 1 |
| calc |
| _ = K*((((2:ℝ)^(g+j+1))^q)*(1/(2:ℝ)^(g+j))^q) := by ring |
| _ = K*(2:ℝ)^q := by |
| rw [dyadic_potential_weight,sub_self,Real.rpow_zero,mul_one] |
| calc |
| _ ≤ ENNReal.ofReal K+∑ j ∈ Finset.range k, ENNReal.ofReal (K*(2:ℝ)^q) := |
| add_le_add hstart (Finset.sum_le_sum hterm) |
| _ = _ := by simp |
|
|
| end Falconer |
|
|
| end OAI |
|
|
