Weak and Strong type $ A_p$ Estimates for Calder\'on-Zygmund Operators

TL;DR

This paper establishes sharp weak and strong type weighted estimates for Calderón-Zygmund operators with smooth kernels, showing bounds depend linearly on the A_p characteristic of the weight for p in (1,2].

Contribution

It proves new sharp weighted bounds for Calderón-Zygmund operators in the range 1<p≤2, linking weak and strong type estimates to the A_p characteristic.

Findings

Weak (L^p(w) to weak-L^p(w)) bounds with A_p dependence for p in (1,2].

Strong (L^2(w) to L^2(w)) bounds with A_2 dependence, combining recent results.

Results are sharp aside from kernel smoothness requirements.

Abstract

For a Calderon-Zygmund operator T on d-dimensional space, that has a sufficiently smooth kernel, we prove that for any 1< p \le 2, and weight w in A_p, that the maximal truncations T_* of T map L^p(w) to weak-L^p(w), with norm bounded by the A_p characteristic of w to the first power. This result combined with the (deep) recent result of Perez-Treil-Volberg, shows that the strong-type of T on L^2(w) is bounded by A_2 characteristic of w to the first power. (It is well-known that L^2 is the critical case for the strong type estimate.) Both results are sharp, aside from the number of derivatives imposed on the kernel of the operator. The proof uses the full structure theory of Calderon-Zygmund Operators, reduction to testing conditions, and a Corona argument.