Interpolating Sequences for the Bergman Space and the ∂̄-equation in Weighted L
نویسنده
چکیده
The author has previously shown that a sequence in the unit disk is a zero sequence for the Bergman space A if and only if a certain weighted L space contains a non-zero (equivalently, zero-free) analytic function. The weight in question is given by a simple formula summed over the zero set. Here we show that a sequence in the unit disk is an interpolating sequence for A if and only if it is separated in the hyperbolic metric and the ∂̄-equation (1−|z|)∂̄u = f has a solution u in this weighted L space whenever f belongs to it. This holds even for p < 1, if the definition of the L space is slightly modified. We provide a proof almost ab initio, constructing a solution operator out of functions whose existence depends on rather basic properties of an interpolation sequence. In particular, this proof does not use the density criterion of K. Seip for interpolation, nor any criterion for weighted ∂̄ estimates. However, we also provide a proof based on Seip’s criterion and a recent criterion for weighted ∂̄ estimates by J. Ortega-Cerdà.
منابع مشابه
Weighted composition operators on weighted Bergman spaces and weighted Bloch spaces
In this paper, we characterize the bonudedness and compactness of weighted composition operators from weighted Bergman spaces to weighted Bloch spaces. Also, we investigate weighted composition operators on weighted Bergman spaces and extend the obtained results in the unit ball of $mathbb{C}^n$.
متن کاملGeneral Interpolating Sequences for the Bergman Spaces
Most characterizations of interpolating sequences for Bergman spaces include the condition that the sequence be uniformly discrete in the hyperbolic metric. We show that if the notion of interpolation is suitably generalized, two of these characterizations remain valid without that condition. The general interpolation we consider here includes the usual simple interpolation and multiple interpo...
متن کاملSelf-commutators of composition operators with monomial symbols on the Bergman space
Let $varphi(z)=z^m, z in mathbb{U}$, for some positive integer $m$, and $C_varphi$ be the composition operator on the Bergman space $mathcal{A}^2$ induced by $varphi$. In this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators $C^*_varphi C_varphi, C_varphi C^*_varphi$ as well as self-commutator and anti-self-commutators of $C_...
متن کاملA new method for the generalized Hyers-Ulam-Rassias stability
We propose a new method, called the textit{the weighted space method}, for the study of the generalized Hyers-Ulam-Rassias stability. We use this method for a nonlinear functional equation, for Volterra and Fredholm integral operators.
متن کامل