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Weighted norm inequalities for Calderon-Zygmund operators without doubling conditions

Xavier Tolsa
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Abstract

In this paper we develop a kind of A_p theory for Calderon-Zygmund operators in a non-homogeneous setting. Let μbe a Borel measure on \R^d which may be non doubling. The only condition that μmust satisfy is μ(B(x,r))\leq Cr^n for all x\in\R^d, r>0 and for some fixed n with 0<n\leq d. We introduce a maximal operator N, which coincides with the maximal Hardy-Littlewood operator if μ(B(x,r))\approx r^n for x\in\supp(μ), and we show that all n-dimensional Calderon-Zygmund operators are bounded on L^p(w dμ) if and only if N is bounded on L^p(w dμ), for a fixed p\in(1,\infty). Also, we prove that this happens if and only if some conditions of Sawyer type hold. This type of weights do not satisfy a reverse Holder inequality, in general, but some kind of self improving property still holds.

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Xavier Tolsa. 2011-10-17. Weighted norm inequalities for Calderon-Zygmund operators without doubling conditions. https://arxiv.org/abs/math/0101192

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