  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 $L^p_\nu$ to $L^p_\eta$ 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
$\sigma(r)\asymp\int_{r}^1\sigma(t)\,dt/(1-r)$ for all $0\leq r<1$. 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.
