Two weight inequality for Bergman projection

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

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