One weight inequality for Bergman projection and Calder\'on operator induced by radial weight

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 $A_p(\omega,\nu)<\infty$.