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

TL;DR

This paper establishes necessary and sufficient conditions for the boundedness of the Bergman projection and Calderón operator on weighted spaces with radial weights, extending classical results to broader weight classes.

Contribution

It introduces a Muckenhoupt-type condition for radial weights that characterizes the boundedness of these operators, generalizing the Forelli-Rudin theorem.

Findings

The Muckenhoupt-type condition is necessary for the one weight inequality.

The condition is sufficient when the weight  is of a specific form.

The Calderf3n operator is bounded if and only if the condition holds.

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