File size: 2,431 Bytes
dae740b | 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 59 60 61 62 63 64 | import Mathlib
namespace OAI
open MeasureTheory Set
open scoped ENNReal NNReal Topology ContDiff
noncomputable section
namespace CenteredDiskEndpoint
/-- The Euclidean plane, not the product space with its max norm. -/
abbrev Plane := EuclideanSpace ℝ (Fin 2)
/-- Absolute disk average, retaining possible infinite values. -/
def absoluteDiskAverage (f : Plane → ℝ) (x : Plane) (r : ℝ) : ℝ≥0∞ :=
(∫⁻ y in Metric.ball x r, ENNReal.ofReal |f y|) /
ENNReal.ofReal (Real.pi * r ^ 2)
/-- The actual centered disk maximal function, over every positive real radius. -/
def maximal (f : Plane → ℝ) (x : Plane) : ℝ≥0∞ :=
⨆ (r : ℝ) (_ : 0 < r), absoluteDiskAverage f x r
/-- Real representative; the endpoint also asserts that `maximal f` is finite a.e. -/
def maximalReal (f : Plane → ℝ) (x : Plane) : ℝ := (maximal f x).toReal
/-- The i-th coordinate derivative of a smooth test function. -/
def testPartial (φ : Plane → ℝ) (i : Fin 2) (x : Plane) : ℝ :=
fderiv ℝ φ x (EuclideanSpace.single i 1)
/-- Distributional first derivatives represented by a vector-valued function. -/
def HasWeakGradient (f : Plane → ℝ) (g : Plane → Plane) : Prop :=
∀ (φ : Plane → ℝ), ContDiff ℝ ∞ φ → HasCompactSupport φ →
∀ i : Fin 2, (∫ x, f x * testPartial φ i x) = -∫ x, g x i * φ x
/-- Local W^(1,1), expressed using the same distributional gradient convention. -/
def InW11Loc (f : Plane → ℝ) : Prop :=
LocallyIntegrable f ∧
∃ g : Plane → Plane, LocallyIntegrable g ∧ HasWeakGradient f g
/-- The gradient L1 norm in Euclidean norm. -/
def gradientNorm (g : Plane → Plane) : ℝ := ∫ x, ‖g x‖
/-- Exact endpoint, including finiteness before passing to the real representative.
Quantifying the input gradient makes the norm bound independent of a choice of
representative of its weak derivatives. -/
def EndpointStatement : Prop :=
∃ C : ℝ, 0 ≤ C ∧
∀ (f : Plane → ℝ) (g : Plane → Plane),
Integrable f → Integrable g → HasWeakGradient f g →
(∀ᵐ x, maximal f x ≠ ⊤) ∧
LocallyIntegrable (maximalReal f) ∧
InW11Loc (maximalReal f) ∧
∃ G : Plane → Plane, Integrable G ∧
HasWeakGradient (maximalReal f) G ∧ gradientNorm G ≤ C * gradientNorm g
theorem main_endpoint : EndpointStatement := by
sorry
end CenteredDiskEndpoint
end
end OAI
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