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Abstract

The purposes of this paper are twofold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil, and Volberg to handle "Bergman-type" singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for the Besov-Sobolev space of analytic functions $B^\sigma_2$ on the complex ball of ${\Bbb{C}}^d$. In particular, we demonstrate that for any $\sigma> 0$, the Carleson measures for the space are characterized by a "T1 Condition". The method of proof of these results is an extension and another application of the work originated by Nazarov, Treil, and the first author.

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