The Reciprocal Problem on Weighted Bergman Spaces

Abstract

We study the reciprocal problem for weighted Bergman spaces: if $f\in A_\alpha^p(\Bn)$ and $\inf_{\Bn}|f|>0$, does it follow that $1/f\in A_\alpha^p(\Bn)$? We determine the range of parameters for which the answer is affirmative. In particular, we prove that functions in $A_{\alpha}^p(\D)$ have the reciprocal property for all $\alpha \in \R$ and $p\geq 1,$ where $\D$ denotes the unit disk in $\C.$ Moreover, functions in $A_{\alpha}^p(\Bn)$ have the reciprocal property for all $\alpha \in \R$ when $n=2$ and $p=2.$ In addition, we resolve the reciprocal problem in the three-dimensional Drury--Arveson space $H_3^2$ and obtain an equivalent condition in the four-dimensional space $H_4^2$. We also settle the reciprocal problem for general $A_{\alpha}^p(\Bn)$ under some additional conditions.