Two weight inequality for Bergman projection

Abstract

The motivation of this paper comes from the two weight inequality $$\Vert P_{\omega }(f){\Vert }_{L_v^p}\le C\Vert f{\Vert }_{L_v^p},f\in L_v^p,$$ for the Bergman projection $P_{\omega }$ in the unit disc. We show that the boundedness of $P_{\omega }$ on $L_v^p$ is characterized in terms of self-improving Muckenhoupt and Bekollé-Bonami type conditions when the radial weights $v$ and $\omega$ admit certain smoothness. En route to the proof we describe the asymptotic behavior of the $L^p$-means and the $L_v^p$-integrability of the reproducing kernels of the weighted Bergman space $A_{\omega }^2$.