Bergman projection on Lebesgue space induced by doubling weight

Abstract

Let $\omega$ and $\nu$ be radial weights on the unit disc of the complex plane, and denote $\sigma ={\omega }^{p^{\prime }}{\nu }^{-\frac{p^{\prime }}{p}}$ 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_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad 1<p<\infty,\tag† \end{equation} for the Bergman projection $P_{\omega }$ induced by $\omega$. It is shown that the moment condition $$D_p(\omega ,\nu )=\underset{n\in \mathbb{N}\cup \{0\}}{\sup }\frac{{\left({\nu }_{np+1}\right)}^{\frac{1}{p}}{\left({\sigma }_{n{p}^{\prime }+1}\right)}^{\frac{1}{p^{\prime }}}}{{\omega }_{2n+1}}<\infty$$ 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 ${\sup }_{0\le r<1}\frac{{\int }_r^1\omega (s)sds}{{\int }_{\frac{1+r}{2}}^1\omega (s)sds}<\infty$ and ${\sup }_{0\le r<1}\frac{{\int }_r^1\omega (s)sds}{{\int }_r^{1-\frac{1-r}{K}}\omega (s)sds}<\infty$ for some $K>1$. In addition, an analogous result for the one weight inequality $\Vert P_{\omega }(f){\Vert }_{D_{\nu ,k}^p}\le C\Vert f{\Vert }_{L_{\nu }^p},$ where $$\Vert f{\Vert }_{D_{\nu ,k}^p}^p=\sum _{j=0}^{k-1}|f^{(j)}(0){|}^p+{\int }_{\mathbb{D}}|f^{(k)}(z){|}^p(1-|z|{)}^{kp}\nu (z)dA(z)<\infty ,k\in \mathbb{N},$$ 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 $\omega (r)=\exp \left(-\frac{\tilde{\alpha }}{(1-r^{\tilde{l}}{)}^{\tilde{\beta }}}\right)$, where $0<\alpha ,\tilde{\alpha },l,\tilde{l}<\infty$ and $0<\beta ,\tilde{\beta }\le 1$.