Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball

Abstract

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 B σ 2 on the complex ball of C d . 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.

Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces

“…So even this is compatible with our theory. Volberg and Wick conclude their paper [VW09] with essentially the same remark, with which Nazarov, Treil and Volberg started theirs [NTV03], that "these considerations can be extended to the case of metric spaces." And indeed they can!…”

“…In the final section, we describe the relation to the above-mentioned results of Volberg and Wick [VW09].…”

“…While the original motivation behind the notion of upper doubling in [Hyt09] was to find a natural unified framework for the doubling and power bounded theories, it also encapsulates other examples of interest. Indeed, although it was not our specific goal, this general framework allows to essentially recapture the recent T 1 theorem of Volberg and Wick [VW09] for "Bergman-type" operators.…”

“…In [VW09] it is checked that this kernel K, the quasi-metric d, and the set H indeed satisfy the Bergman-type kernel estimates. We now observe that even the standard estimates of our theory are verified.…”

Fractional derivatives of pointwise multipliers between holomorphic spaces

“…For general s > 0, we recall, for instance, that if n − sp < 1, the Carleson measures for the spaces H p s can be characterized by a capacitary condition on open sets in S (see [18]) and for H 2 s and any s > 0, was obtained (see [29]) a non-capacitary characterization. A non-negative Borel measure µ on S is a trace measure for H …”