Bergman projection on Lebesgue space induced by doubling weight

Let $\omega$ and $\nu$ be radial weights on the unit disc of the complex plane, and denote $\sigma=\omega^{p'}\nu^{-\frac{p'}p}$ and $\omega_x =\int_0^1 s^x \omega(s)\,ds$ for all $1\le x<\infty$. 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 moment condition $$D_p(\omega,\nu)=\sup_{n\in \mathbb{N}\cup\{0\}}\frac{\left(\nu_{np+1}\right)^\frac1p\left(\sigma_{np'+1}\right)^\frac1{p'}}{\omega_{2n+1}}<\infty$$ is necessary for \eqref{ab1} to hold. Further, $D_p(\omega,\nu)<\infty$ is also sufficient for \eqref{ab1} if $\nu$ admits the doubling properties $\sup_{0\le r<1}\frac{\int_r^1 \omega(s)s\,ds}{\int_{\frac{1+r}{2}}^1 \omega(s)s\,ds}<\infty$ and $\sup_{0\le r<1}\frac{\int_r^1 \omega(s)s\,ds}{\int_r^{1-\frac{1-r}{K}} \omega(s)s\,ds}<\infty$ for some $K>1$. In addition, an analogous result for the one weight inequality $\|P_\omega(f)\|_{D^p_{\nu,k}}\le C\|f\|_{L^p_\nu},$ where $$\Vert f \Vert_{D^p_{\nu, k}}^p =\sum\limits_{j=0}^{k-1}| f^{(j)}(0)|^p+ \int_{\mathbb{D}} \vert f^{(k)}(z)\vert^p (1-|z| )^{kp}\nu(z)\,dA(z)<\infty, \quad k\in \mathbb{N},$$ is established. The inequality \eqref{ab1} is further studied by using the necessary condition $D_p(\omega,\nu)<\infty$ in the case of the exponential type weights $\nu(r)=\exp \left(-\frac{\alpha}{(1-r^l)^{\beta}} \right)$ and $\omega(r)= \exp \left(-\frac{\widetilde{\alpha}}{(1-r^{\widetilde{l}})^{\widetilde{\beta}}} \right)$, where $0<\alpha, \, \widetilde{\alpha}, \, l, \, \widetilde{l}<\infty$ and $0<\beta , \, \widetilde{\beta}\le 1$.