An Integral Formula in Weighted Bergman Spaces

Abstract

In order to establish that extremal functions in the Bergman space A p act as both expansive multipliers and contractive divisors, Duren, Khavinson, Shapiro and Sundberg made use of an integral formula involving the biharmonic Green function. Using a weighted biharmonic Green function, we derive an analogous integral formula in the standard weighted Bergman space \(A_{\alpha}^{p}\ {\rm when}\ \alpha =1\), and we also discuss how the formula can be established for general α. Moreover, we show that each \(A_{\alpha}^{p}\) inner function acts as a contractive divisor on the invariant subspace which it generates.