File size: 1,471 Bytes
cfa2fd2 | 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 54 55 56 57 58 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal NNReal
namespace TriangularHilbert
abbrev Plane := ℝ × ℝ
structure Endpoints where
lower : ℝ
upper : ℝ
lower_pos : 0 < lower
lower_lt_upper : lower < upper
def annulus (ε R : ℝ) : Set ℝ := {t | ε < |t| ∧ |t| < R}
def integrand (F G : Plane → ℂ) (z : Plane) (t : ℝ) : ℂ :=
F (z.1 + t, z.2) * G (z.1, z.2 + t) / (t : ℂ)
def truncation (F G : Plane → ℂ) (E : Endpoints) (z : Plane) : ℂ :=
∫ t in annulus E.lower E.upper, integrand F G z t
def GoodPoint (F G : Plane → ℂ) (z : Plane) : Prop :=
∀ n : ℕ, IntegrableOn (integrand F G z)
(annulus (1 / ((n : ℝ) + 2)) ((n : ℝ) + 2))
def maximal (F G : Plane → ℂ) (z : Plane) : ℝ≥0∞ := by
classical
exact if GoodPoint F G z then
⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0
def maximalNorm (F G : Plane → ℂ) : ℝ≥0∞ :=
(∫⁻ z : Plane, maximal F G z ^ ((3 : ℝ) / 2)) ^ ((2 : ℝ) / 3)
def MainEstimate : Prop :=
∃ C : ℝ≥0, ∀ F G : Plane → ℂ,
MemLp F 3 volume → MemLp G 3 volume →
(∀ᵐ z ∂volume, GoodPoint F G z) ∧
AEMeasurable (maximal F G) volume ∧
maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volume
open MeasureTheory Filter
open scoped ENNReal
theorem main_estimate : MainEstimate := by
sorry
end TriangularHilbert
end
end OAI
|