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465b10d | 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 | import Mathlib
namespace OAI
noncomputable section
namespace SignedDisk
open MeasureTheory Set
open scoped ENNReal NNReal Topology ContDiff
namespace CenteredDiskEndpoint
/-- The Euclidean plane, with its Euclidean norm. -/
abbrev Plane := EuclideanSpace ℝ (Fin 2)
def diskAverage (f : Plane → ℝ) (x : Plane) (r : ℝ) : ℝ :=
(∫ y in Metric.ball x r, f y) / (Real.pi * r ^ 2)
/-- 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
/-- The gradient L1 norm in Euclidean norm. -/
def gradientNorm (g : Plane → Plane) : ℝ := ∫ x, ‖g x‖
/-- Signed envelope over all real radii in the closed band, including its endpoints. -/
def signedBand (f : Plane → ℝ) (a b : ℝ) (x : Plane) : ℝ :=
sSup ((fun r => diskAverage f x r) '' Icc a b)
def SignedFiniteBandStatement : Prop :=
∃ C : ℝ, 0 ≤ C ∧
∀ (f : Plane → ℝ), ContDiff ℝ ∞ f → HasCompactSupport f →
∀ (a : ℝ), 0 < a → ∀ m : ℕ,
LocallyIntegrable (signedBand f a ((2 : ℝ)^m * a)) ∧
∃ G : Plane → Plane, Integrable G ∧
HasWeakGradient (signedBand f a ((2 : ℝ)^m * a)) G ∧
gradientNorm G ≤ C * gradientNorm (gradient f)
theorem main_signed_finite_band : SignedFiniteBandStatement := by
sorry
end CenteredDiskEndpoint
end SignedDisk
end
end OAI
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