| import OAI.MeasureTheory.Falconer.Radial.RadialScaleWindow |
| import OAI.MeasureTheory.Falconer.Recursion.RelativeScaleBudget |
| import OAI.MeasureTheory.Falconer.Radial.RadialDyadicBracket |
|
|
| namespace OAI |
|
|
| /-! # From a window of improved dyadic caps to every radius -/ |
|
|
| namespace Falconer |
| open scoped ENNReal |
|
|
| theorem dyadic_window_interpolation (f : ℝ → ℝ≥0∞) (hf : Monotone f) |
| (hmass : ∀ r : ℝ, f r ≤ 1) {N m : ℕ} (_hmN : m ≤ N) |
| {e q v : ℝ} (hq : 0 ≤ q) (hqv : q ≤ v) |
| (hcoarse : (m:ℝ)*q ≤ (N:ℝ)*e) |
| (hconstant : (2:ℝ)^v ≤ ((2:ℝ)^N)^e) |
| (hdyadic : ∀ j : ℕ, m ≤ j → j ≤ N → f (1/(2:ℝ)^j) ≤ |
| ENNReal.ofReal ((1/(2:ℝ)^j)^v)) |
| {r : ℝ} (hr : 1/(2:ℝ)^N ≤ r) (hr1 : r ≤ 1) : |
| f r ≤ ENNReal.ofReal (((2:ℝ)^N)^e*r^q) := by |
| have hr0 : 0 < r := (by positivity : 0 < 1/(2:ℝ)^N).trans_le hr |
| by_cases hc : 1/(2:ℝ)^m ≤ r |
| · apply (hmass r).trans |
| rw [← ENNReal.ofReal_one] |
| apply ENNReal.ofReal_le_ofReal |
| have hp := dyadic_relative_scale_bound m N hcoarse |
| have hi : ((2:ℝ)^m)^q*(1/(2:ℝ)^m)^q = 1 := by |
| rw [one_div,Real.inv_rpow (by positivity)] |
| exact mul_inv_cancel₀ (by positivity) |
| calc |
| 1 = ((2:ℝ)^m)^q*(1/(2:ℝ)^m)^q := hi.symm |
| _ ≤ ((2:ℝ)^N)^e*r^q := mul_le_mul hp |
| (Real.rpow_le_rpow (by positivity) hc hq) (by positivity) (by positivity) |
| · obtain ⟨j,hj,hrj,hjr⟩ := Regularization.dyadic_radius_bracket hr hr1 |
| have hmj : m ≤ j := by |
| by_contra hn |
| have hjm : j+1 ≤ m := by omega |
| have hd : 1/(2:ℝ)^m ≤ 1/(2:ℝ)^(j+1) := |
| one_div_le_one_div_of_le (by positivity) (pow_le_pow_right₀ (by norm_num) hjm) |
| have he : 1/(2:ℝ)^j = 2*(1/(2:ℝ)^(j+1)) := by rw [pow_succ] |
| have hh : 1/(2:ℝ)^(j+1) ≤ r := by rw [he] at hjr |
| exact hc (hd.trans hh) |
| apply ((hf hrj).trans (hdyadic j hmj hj)).trans (ENNReal.ofReal_le_ofReal ?_) |
| calc |
| _ ≤ (2*r)^v := Real.rpow_le_rpow (by positivity) hjr (hq.trans hqv) |
| _ = (2:ℝ)^v*r^v := Real.mul_rpow (by norm_num) hr0.le |
| _ ≤ ((2:ℝ)^N)^e*r^q := mul_le_mul hconstant |
| (Real.rpow_le_rpow_of_exponent_ge hr0 hr1 hqv) (by positivity) (by positivity) |
|
|
| end Falconer |
|
|
| end OAI |
|
|
