Two weight inequality for Bergman projection

The motivation of this paper comes from the two weight inequality $$\|P_\omega(f)\|_{L^p_v}\le C\|f\|_{L^p_v},\quad f\in L^p_v,$$ for the Bergman projection $P_\omega$ in the unit disc. We show that the boundedness of $P_\omega$ on $L^p_v$ is characterized in terms of self-improving Muckenhoupt and Bekoll\'e-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 $L^p_v$-integrability of the reproducing kernels of the weighted Bergman space $A^2_\omega$.