Weak and Strong type $ A_p$ Estimates for Calderón-Zygmund Operators
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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.
