Muckenhoupt type weights and Berezin formulas for Bergman spaces

Abstract

By means of Muckenhoupt type conditions, we characterize the weights $\omega$ on $\C$ such that the Bergman projection of $F^{2,\ell}_{\alpha}=H(\C)\cap L^2(\C,e^{-\frac{\alpha}2|z|^{2\ell}})$, $\alpha>0$, $\ell>1$, is bounded on $L^p(\C,e^{-\frac{\alpha p}2|z|^{2\ell}}\omega(z))$, for $1<p<\infty$. We also obtain explicit representation integral formulas for functions in the weighted Bergman spaces $A^p(\omega)=H(\C)\cap L^p(\omega)$. Finally, we check the validity of the so called Sarason conjecture about the boundedness of products of certain Toeplitz operators on the spaces $F^{p,\ell}_\alpha=H(\C)\cap L^p(\C,e^{-\frac{\alpha p}2|z|^{2\ell}})$.