| import Mathlib |
|
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| namespace OAI |
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| namespace RieszRectifiability |
| noncomputable section |
| open MeasureTheory Set Metric |
| open scoped NNReal ENNReal |
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| abbrev Ambient (d : ℕ) := EuclideanSpace ℝ (Fin d) |
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| def AdmissibleRadius {d : ℕ} (μ : Measure (Ambient d)) (r : ℝ) : Prop := |
| 0 < r ∧ ENNReal.ofReal r ≤ Metric.ediam μ.support |
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| def kernel {d : ℕ} (n : ℕ) (x y : Ambient d) : Ambient d := |
| (‖x - y‖ ^ (n + 1))⁻¹ • (x - y) |
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| def truncated {d : ℕ} (n : ℕ) (μ : Measure (Ambient d)) |
| (ε : ℝ) (f : Ambient d → ℝ) (x : Ambient d) : Ambient d := |
| ∫ y in {y | ε < dist x y}, f y • kernel n x y ∂μ |
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| def ADRegularWithConstant {d : ℕ} (n : ℕ) (C : ℝ) |
| (μ : Measure (Ambient d)) : Prop := |
| 1 ≤ C ∧ ∀ x ∈ μ.support, ∀ r : ℝ, AdmissibleRadius μ r → |
| ENNReal.ofReal (r ^ n / C) ≤ μ (ball x r) ∧ |
| μ (ball x r) ≤ ENNReal.ofReal (C * r ^ n) |
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| def RieszL2BoundedWithConstant {d : ℕ} (n : ℕ) (C : ℝ≥0) |
| (μ : Measure (Ambient d)) : Prop := |
| ∀ ε : ℝ, 0 < ε → ∀ f : Ambient d → ℝ, |
| MemLp f 2 μ → MemLp (truncated n μ ε f) 2 μ ∧ |
| eLpNorm (truncated n μ ε f) 2 μ ≤ (C : ℝ≥0∞) * eLpNorm f 2 μ |
|
|
| def BallImageConclusion {d : ℕ} (n : ℕ) (μ : Measure (Ambient d)) |
| (θ : ℝ) (M : ℝ≥0) : Prop := |
| ∀ x ∈ μ.support, ∀ r : ℝ, AdmissibleRadius μ r → |
| ∃ g : (ball (0 : Ambient n) r) → Ambient d, |
| LipschitzWith M g ∧ |
| ENNReal.ofReal (θ * r ^ n) ≤ μ (ball x r ∩ range g) |
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| / |
| def QuantitativeFullStatement : Prop := |
| ∀ d n : ℕ, 4 ≤ d → 2 ≤ n → n + 2 ≤ d → |
| ∀ C_AD : ℝ, 1 ≤ C_AD → ∀ C_R : ℝ≥0, |
| ∃ θ : ℝ, 0 < θ ∧ ∃ M : ℝ≥0, |
| ∀ μ : Measure (Ambient d), μ.Regular → |
| ADRegularWithConstant n C_AD μ → |
| RieszL2BoundedWithConstant n C_R μ → |
| BallImageConclusion n μ θ M |
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|
| end |
| end RieszRectifiability |
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| namespace RieszRectifiability |
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| / |
| theorem quantitative_higher_codimension_riesz_rectifiability : QuantitativeFullStatement := by |
| sorry |
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| end RieszRectifiability |
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| end OAI |
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|
