Thin Sequences and Their Role in Model Spaces and Douglas Algebras

We study thin interpolating sequences ${lambda_n}$ and their relationship to interpolation in the Hardy space $H^2$ and the model spaces $K_Theta = H^2 ominus Theta H^2$, where $Theta$ is an inner function. Our results, phrased in terms of the functions that do the interpolation as well as Carleson measures, show that under the assumption that $Theta(lambda_n) to 0$ the interpolation properties in $H^2$ are essentially the same as those in $K_Theta$.

On the uniqueness sets in the Fock space

It was known to von Neumann in the 1950s that the integer lattice $mathbb{Z}^2$ forms a uniqueness set for the Bargmann-Fock space. It was later demonstrated by Seip and Wallsten that a sequence of points $Gamma$ that is uniformly close to the integer lattice is still a uniqueness set. We show in this paper that the uniqueness sets for the Fock space are preserved under much more general perturbations.

Localization and Compactness in Bergman and Fock spaces

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when $1 < p < infty$, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the $1 < p < infty$ case.

Bergman-type Singular Operators and the Characterization of Carleson Measures for Besov--Sobolev Spaces on the Complex Ball

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 $B^sigma_2$ on the complex ball of $mathbb{C}^d$. 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.

Toeplitz Corona Theorems for the Polydisk and the Unit Ball

The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in $mathbb{C}^n$.