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One weight inequality for Bergman projection and Calderón operator induced by radial weight
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by Francisco J. Martín Reyes, Pedro Ortega, José Ángel Peláez and Jouni Rättyä;

Proc. Amer. Math. Soc. 151 (2023), 5189-5203

DOI: https://doi.org/10.1090/proc/15831

Published electronically: September 21, 2023

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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} \|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 \begin{equation*} 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)^{\frac 1p}}\right )^{p’}\,ds \right )^{\frac {1}{p’}}}{\int _r^1 s\omega (s)\,ds}<\infty \end{equation*} is necessary for ($\dag$) to hold, and sufficient if $\nu$ is of the form \begin{align*} \nu (s)=\omega (s)\left (\int _r^1 s\omega (s)\,ds \right )^\alpha \end{align*} 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 \begin{equation*} 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} \end{equation*} is bounded on $L^p_\nu$ if and only if $A_p(\omega ,\nu )<\infty$.

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