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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 49 50 51 52 53 | 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]; field_simp
have hh : 1/(2:ℝ)^(j+1) ≤ r := by rw [he] at hjr; linarith only [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
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