Radial two weight inequality for maximal Bergman projection induced by a regular weight

It is shown in quantitative terms that the maximal Bergman projection \begin{equation*} P^{+}\omega(f)(z)=\int$\mathbb{D}$ f(\zeta)|B^\omega_z(\zeta)|\omega(\zeta),dA(\zeta), \end{equation*} is bounded from $L_{\nu }^p$ to $L_{\eta }^p$ if and only if \begin{equation*} \sup_{0<r<1}\left(\int_0^r\frac{\eta(s)}{\left(\int_{s}^1\omega(t),dt\right)^p},ds\right)^{\frac{1}{p}} \left(\int_r^1\left(\frac{\omega(s)}{\nu(s)^\frac{1}{p}}\right)^{p’}ds\right)^{\frac{1}{p’}}<\infty, \end{equation*} provided $\omega ,\nu ,\eta$ are radial regular weights. A radial weight $\sigma$ is regular if it satisfies $\sigma (r)\asymp {\int }_r^1\sigma (t)dt\mathrm{/}(1-r)$ for all $0\le r<1$. It is also shown that under an appropriate additional hypothesis involving $\omega$ and $\eta$, the Bergman projection $P_{\omega }$ and $P_{\omega }^{+}$ are simultaneously bounded.