| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory |
| open scoped ENNReal NNReal |
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| namespace TriangularHilbert |
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| abbrev Plane := ℝ × ℝ |
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| structure Endpoints where |
| lower : ℝ |
| upper : ℝ |
| lower_pos : 0 < lower |
| lower_lt_upper : lower < upper |
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| def annulus (ε R : ℝ) : Set ℝ := {t | ε < |t| ∧ |t| < R} |
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| 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) |
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| def truncation (F G : Plane → ℂ) (E : Endpoints) (z : Plane) : ℂ := |
| ∫ t in annulus E.lower E.upper, integrand F G z t |
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| def GoodPoint (F G : Plane → ℂ) (z : Plane) : Prop := |
| ∀ n : ℕ, IntegrableOn (integrand F G z) |
| (annulus (1 / ((n : ℝ) + 2)) ((n : ℝ) + 2)) |
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| 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 |
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| def maximalNorm (F G : Plane → ℂ) : ℝ≥0∞ := |
| (∫⁻ z : Plane, maximal F G z ^ ((3 : ℝ) / 2)) ^ ((2 : ℝ) / 3) |
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| 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 |
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| open MeasureTheory Filter |
| open scoped ENNReal |
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| theorem main_estimate : MainEstimate := by |
| sorry |
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| end TriangularHilbert |
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| end |
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| end OAI |
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