The purposes of this paper are two fold. 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 B2σ​ on the complex ball of Cd. In particular, we
demonstrate that for any σ>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.Comment: v1: 31 pgs; v2: 31 pgs, title changed, typos corrected, references
added; v3: 33 pages, typos corrected, references added, presentation improved
based on referee comments