One weight inequality for Bergman projection and Calderón operator induced by radial weight

Abstract

Let $ω$ and $ν$ be radial weights on the unit disc of the complex plane such that $ω$ admits the doubling property $\sup_{0\le r<1}\frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1 ω(s)\,ds}<\infty$. 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 $P_ω$ induced by $ω$. It is shown that the Muckenhoupt-type condition $$ A_p(ω,ν)=\sup_{0\le r<1}\frac{\left(\int_r^1 sν(s)\,ds \right)^{\frac{1}{p}}\left(\int_r^1 s\left(\frac{ω(s)}{ν(s)^{\frac1p}}\right)^{p'}\,ds \right)^{\frac{1}{p'}}}{\int_r^1 sω(s)\,ds}<\infty, $$ is necessary for \eqref{ab1} to hold, and sufficient if $ν$ is of the form $ν(s)=ω(s)\left(\int_r^1 sω(s)\,ds \right)^α$ for some $-1<α<\infty$. 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 $(ω,ν)$ of radial weights the Calderón operator $$ H^\star_ω(f)(z)+H_ω(f)(z) =\int_{0}^{|z|} f\left(s\frac{z}{|z|}\right)\frac{sω(s)\,ds}{\int_s^1 tω(t)\,dt} +\frac{\int_{|z|}^1f\left(s\frac{z}{|z|}\right) sω(s)\,ds}{\int_{|z|}^1 sω(s)\,ds}\,ds $$ is bounded on $L^p_ν$ if and only if $A_p(ω,ν)<\infty$.