| import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp |
| import Mathlib.MeasureTheory.Function.L2Space |
|
|
| namespace OAI |
|
|
| /-! # Positive powers of a bounded product of two L2 functions -/ |
|
|
| open Set Filter MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} {f g : X → ℝ} |
|
|
| theorem bounded_product_power_integrable (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) |
| {A B : ℝ} (ha : ∀ᵐ x ∂μ, |f x| ≤ A) (hb : ∀ᵐ x ∂μ, |g x| ≤ B) |
| (hA : 0 ≤ A) (_hB : 0 ≤ B) {q : ℕ} (hq : 1 ≤ q) : |
| Integrable (fun x => (f x*g x)^q) μ := by |
| have hi : Integrable (fun x => f x*g x) μ := hf.integrable_mul hg |
| apply (hi.norm.const_mul ((A*B)^(q-1))).mono' ((hf.aestronglyMeasurable.mul hg.aestronglyMeasurable).pow q) |
| filter_upwards [ha,hb] with x hax hbx |
| have hbound : |f x*g x| ≤ A*B := by |
| rw [abs_mul] |
| exact mul_le_mul hax hbx (abs_nonneg _) hA |
| change ‖(f x*g x)^q‖ ≤ (A*B)^(q-1)*‖f x*g x‖ |
| simp only [Real.norm_eq_abs,abs_pow] |
| calc |
| |f x*g x|^q = |f x*g x|^(q-1)*|f x*g x| := by |
| rw [← pow_succ] |
| congr 1 |
| omega |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (pow_le_pow_left₀ (abs_nonneg _) hbound (q-1)) (abs_nonneg _) |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
