  Let $\omega$ and $\nu$ be radial weights on the unit disc of the complex
plane such that $\omega$ admits the doubling property $\sup_{0\le
r<1}\frac{\int_r^1 \omega(s)\,ds}{\int_{\frac{1+r}{2}}^1
\omega(s)\,ds}<\infty$. Consider the one weight inequality
  \begin{equation}\label{ab1}
  \|P_\omega(f)\|_{L^p_\nu}\le C\|f\|_{L^p_\nu},\quad 1<p<\infty,\tag{\dag}
  \end{equation} for the Bergman projection $P_\omega$ induced by $\omega$. It
is shown that the Muckenhoupt-type condition
  $$
  A_p(\omega,\nu)=\sup_{0\le r<1}\frac{\left(\int_r^1 s\nu(s)\,ds
\right)^{\frac{1}{p}}\left(\int_r^1
s\left(\frac{\omega(s)}{\nu(s)^{\frac1p}}\right)^{p'}\,ds
\right)^{\frac{1}{p'}}}{\int_r^1 s\omega(s)\,ds}<\infty,
  $$ is necessary for \eqref{ab1} to hold, and sufficient if $\nu$ is of the
form $\nu(s)=\omega(s)\left(\int_r^1 s\omega(s)\,ds \right)^\alpha$ for some
$-1<\alpha<\infty$. This result extends the classical theorem due to Forelli
and Rudin for a much larger class of weights. In addition, it is shown that for
any pair $(\omega,\nu)$ of radial weights the Calder\'on operator
  $$
  H^\star_\omega(f)(z)+H_\omega(f)(z)
  =\int_{0}^{|z|} f\left(s\frac{z}{|z|}\right)\frac{s\omega(s)\,ds}{\int_s^1
t\omega(t)\,dt}
  +\frac{\int_{|z|}^1f\left(s\frac{z}{|z|}\right) s\omega(s)\,ds}{\int_{|z|}^1
s\omega(s)\,ds}\,ds
  $$ is bounded on $L^p_\nu$ if and only if $A_p(\omega,\nu)<\infty$.
