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

Abstract

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_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 Muckenhoupt-type condition $$A_p(\omega ,\nu )=\underset{0\le r<1}{\sup }\frac{{\left({\int }_r^{1}s\nu (s)ds\right)}^{\frac{1}{p}}{\left({\int }_r^{1}s{\left(\frac{\omega (s)}{\nu (s{)}^{\frac{1}{p}}}\right)}^{p^{\prime }}ds\right)}^{\frac{1}{p^{\prime }}}}{{\int }_r^{1}s\omega (s)ds}<\infty ,$$ 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 Calderón operator $$H_{\omega }^{\star }(f)(z)+H_{\omega }(f)(z)={\int }_0^{|z|}f\left(s\frac{z}{|z|}\right)\frac{s\omega (s)ds}{{\int }_s^{1}t\omega (t)dt}+\frac{{\int }_{|z|}^{1}f\left(s\frac{z}{|z|}\right)s\omega (s)ds}{{\int }_{|z|}^{1}s\omega (s)ds}ds$$ is bounded on $L_{\nu }^p$ if and only if $A_p(\omega ,\nu )<\infty$.