Calderón–Zygmund operators on the full Lᵖ space #
The strong-type estimates for a Calderón–Zygmund operator initially act on L² ∩ Lᵖ.
Finite-measure simple functions are dense in Lᵖ for p < ∞, so these estimates determine a
bounded linear operator on all of Lᵖ. The extension agrees almost everywhere with the original
operator on the intersection. For p > 2, the cancellation hypothesis is applied to the adjoint.
References #
- A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.
- E. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter II, §2.
theorem
ContinuousLinearMap.exists_strongTypeExtension_of_setLIntegral_compl_closedBall_le
{ι : Type u_1}
[Fintype ι]
[Nonempty ι]
{H : Type u_2}
{K : Type u_3}
[NormedAddCommGroup H]
[InnerProductSpace ℝ H]
[CompleteSpace H]
[NormedAddCommGroup K]
[InnerProductSpace ℝ K]
[CompleteSpace K]
{p : ENNReal}
[Fact (1 ≤ p)]
(T : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume) →L[ℝ] ↥(MeasureTheory.Lp K 2 MeasureTheory.volume))
{B B' : ENNReal}
(hB : B ≠ ⊤)
(hB' : B' ≠ ⊤)
(hT :
∀ (b : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ),
(∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) →
∫ (x : ι → ℝ), ↑↑b x = 0 →
∫⁻ (x : ι → ℝ) in (Metric.closedBall y (2 * r))ᶜ, ‖↑↑(T b) x‖ₑ ≤ B * ∫⁻ (x : ι → ℝ), ‖↑↑b x‖ₑ)
(hT' :
∀ (b : ↥(MeasureTheory.Lp K 2 MeasureTheory.volume)) (y : ι → ℝ) (r : ℝ),
(∀ᵐ (x : ι → ℝ), x ∉ Metric.closedBall y r → ↑↑b x = 0) →
∫ (x : ι → ℝ), ↑↑b x = 0 →
∫⁻ (x : ι → ℝ) in (Metric.closedBall y (2 * r))ᶜ, ‖↑↑((adjoint T) b) x‖ₑ ≤ B' * ∫⁻ (x : ι → ℝ), ‖↑↑b x‖ₑ)
(hp : 1 < p)
(hp_top : p ≠ ⊤)
:
∃ (C : NNReal) (S : ↥(MeasureTheory.Lp H p MeasureTheory.volume) →L[ℝ] ↥(MeasureTheory.Lp K p MeasureTheory.volume)),
‖S‖ ≤ ↑C ∧ ∀ (f : ↥(MeasureTheory.Lp H 2 MeasureTheory.volume)) (hf : MeasureTheory.MemLp (↑↑f) p MeasureTheory.volume),
↑↑(S (MeasureTheory.MemLp.toLp (↑↑f) hf)) =ᵐ[MeasureTheory.volume] ↑↑(T f)
The full strong-type Calderón–Zygmund theorem. An L²-bounded operator for which both
it and its adjoint satisfy the cancellation estimate with finite constants extends to a bounded
operator on Lᵖ(ℝⁿ; H) → Lᵖ(ℝⁿ; K) for every 1 < p < ∞. On L² ∩ Lᵖ, the extended map agrees
almost everywhere with the given operator.