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

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

2010-06-13

arxiv.org/abs/1006.2530v3

Mathematics

Classical Analysis and ODEs

This paper has been withdrawn by the authors. The Theorems of this paper are extended and improved in arXiv:1103.5229. The pro

Scientific paper

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.

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