The Fredholm properties of Toeplitz operators on the Bergman space A have been well-known for continuous symbols since the 1970s. We investigate the case p = 1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on A that ari...

The aim of this work is to find optimal estimates for the Bergman kernels classical domains ℜ

Building on [BB08a], we prove a general criterion for convergence of (possibly singular) Bergman measures towards equilibrium measures on complex manifolds. The criterion may be formulated in terms of growth properties of balls of holomorphic sections, or equivalently as an asymptotic minimization of generalized Donaldson L-functionals. Our result yields in particular the proof of a well-known ...

Received 13 June 2008; Revised 12 October 2008; Accepted 20 November 2008 Recommended by Stevo Stevic This paper shows that if S is a bounded linear operator acting on the weighted Bergman spaces Aα on the unit ball in C n such that STzi TziS i 1, . . . , n , where Tzi zif and Tzi P zif ; and where P is the weighted Bergman projection, then S must be a Hankel operator. Copyright q 2008 Y. Lu an...

We present an overview of uniformly discrete sequences and their relationship to other important types of sets, such as uniformly separated sequences, sets of interpolation and sampling, and zero-sets for Bergman spaces. Introduction A sequence Γ = {zk} in the unit disk D is said to be uniformly discrete if ρ(zj , zk) = ∣∣∣∣ zj − zk 1− zjzk ∣∣∣∣ ≥ δ for some fixed δ > 0 and all j 6= k. The supr...

In this paper, we study general extremal problems for non-vanishing functions in Bergman spaces. We show the existence and uniqueness of solutions to a wide class of such problems. In addition, we prove certain regularity results: the extremal functions in the problems considered must be in a Hardy space, and in fact must be bounded. We conjecture what the exact form of the extremal function is...

In this paper we present an abstract framework for construction of Banach spaces of distributions from group representations. This generalizes the theory of coorbit spaces initiated by H.G. Feichtinger and K. Gröchenig in the 1980’s. Spaces that can be described by this new technique include the whole Banach-scale of Bergman spaces on the unit disc. For these Bergman spaces we show that atomic ...

The contents of this work are part of the author’s doctoral dissertation, which was conducted under the supervision of Prof. George M. Bergman, at the University of California at Berkeley. It is my very great pleasure to express my deep gratitude and indebtedness to Prof. Bergman, for his patience and his every-ready help and advice. The formulation in Section 2 is due to him, as are the combin...

We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functi...