-
Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 4, August 2012
- pp. 949-992
- 10.1353/ajm.2012.0028
- Article
- Additional Information
- Purchase/rental options available:
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.