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0789dda | 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import Mathlib
namespace OAI
noncomputable section
/- Use the normed-field structure on ℂ for real scalar multiplication. -/
attribute [-instance] instCommCStarAlgebraComplex
open MeasureTheory Set
open scoped ENNReal NNReal
namespace LipschitzHilbert
abbrev Plane := EuclideanSpace ℝ (Fin 2)
abbrev DirectionPlane := EuclideanSpace ℝ (Fin 2)
def sample (v : DirectionPlane → DirectionPlane) (t : ℝ) (x : DirectionPlane) :
DirectionPlane := x - t • v x
/-- The symmetrized form of the hard-truncated directional integral. -/
def shortTransform (v : DirectionPlane → DirectionPlane) (ε a : ℝ)
(f : DirectionPlane → ℂ) (x : DirectionPlane) : ℂ :=
∫ t in ε..a, t⁻¹ • (f (sample v t x) - f (sample v (-t) x))
def directionalKernel (v : DirectionPlane → DirectionPlane)
(f : DirectionPlane → ℂ) (x : DirectionPlane) (t : ℝ) : ℂ :=
t⁻¹ • f (sample v t x)
/-- The two hard-cutoff intervals in the directional Hilbert transform. -/
def twoSidedTruncation (v : DirectionPlane → DirectionPlane) (ε a : ℝ)
(f : DirectionPlane → ℂ) (x : DirectionPlane) : ℂ :=
(∫ t in -a..-ε, directionalKernel v f x t) +
∫ t in ε..a, directionalKernel v f x t
/-- The symmetric quotient extended continuously at zero for smooth inputs. -/
def smoothPrincipalValue (v : DirectionPlane → DirectionPlane) (a : ℝ)
(f : DirectionPlane → ℂ) (x : DirectionPlane) : ℂ :=
let g := fun t : ℝ => f (sample v t x)-f (sample v (-t) x)
∫ t in (0:ℝ)..a, Function.update (fun t => t⁻¹ • g t) 0 (deriv g 0) t
attribute [instance] instCommCStarAlgebraComplex
def ShortHilbertConclusion (a : ℝ) (C : ℝ≥0) : Prop :=
∀ v : Plane → Plane, LipschitzWith 1 v → (∀ x, ‖v x‖=1) →
(∀ epsilon : ℝ, 0<epsilon → epsilon<a → ∀ f : SchwartzMap Plane ℂ,
eLpNorm (twoSidedTruncation v epsilon a f) 2 volume≤C*eLpNorm f 2 volume) ∧
(∀ f : SchwartzMap Plane ℂ,
eLpNorm (smoothPrincipalValue v a f) 2 volume≤C*eLpNorm f 2 volume) ∧
(∀ f : SchwartzMap Plane ℂ, ∀ level : ℝ≥0∞, level≠0 → level≠∞ →
volume {x | level<‖smoothPrincipalValue v a f x‖ₑ}≤
level⁻¹^2*((C:ℝ≥0∞)*eLpNorm f 2 volume)^2) ∧
(∀ epsilon : ℝ, 0<epsilon → epsilon<a →
∃ T : Lp ℂ 2 (volume : Measure Plane) →L[ℂ] Lp ℂ 2 volume,
‖T‖≤C ∧ ∀ f : SchwartzMap Plane ℂ,
(T (f.toLp 2 volume) : Plane → ℂ)=ᵐ[volume] twoSidedTruncation v epsilon a f) ∧
(∃ T : Lp ℂ 2 (volume : Measure Plane) →L[ℂ] Lp ℂ 2 volume,
‖T‖≤C ∧ ∀ f : SchwartzMap Plane ℂ,
(T (f.toLp 2 volume) : Plane → ℂ)=ᵐ[volume] smoothPrincipalValue v a f)
/-- Uniform hard truncation, principal-value and weak-(2,2) estimates,
with bounded L² extensions for all Lipschitz unit vector fields. -/
theorem main :
∃ a : ℝ, a∈Ioo 0 (1/2) ∧ ∃ C : ℝ≥0, 0<C ∧ ShortHilbertConclusion a C := by
sorry
/-- The same uniform estimate at the reciprocal-Lipschitz truncation scale. -/
theorem uniform_truncation :
∃ C : ℝ, 0<C ∧ ∀ K : ℝ≥0, 0<K → ∀ epsilon R : ℝ,
0<epsilon → epsilon≤R → R≤1/(1000000*(K:ℝ)) →
∀ v : Plane → Plane, LipschitzWith K v → (∀ x, ‖v x‖=1) →
∀ f : SchwartzMap Plane ℂ,
eLpNorm (shortTransform v epsilon R f) 2 volume≤ENNReal.ofReal (C*‖f.toLp 2 volume‖) := by
sorry
end LipschitzHilbert
end
end OAI
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