Radial averaging operator acting on Bergman and Lebesgue spaces

Abstract

It is shown that the radial averaging operator $$ T_ω(f)(z)=\frac{\int_{|z|}^1f\left(s\frac{z}{|z|}\right)ω(s)\,ds}{\widehatω(z)},\quad \widehatω(z)=\int_{|z|}^1ω(s)\,ds, $$ induced by a radial weight $ω$ on the unit disc $\mathbb{D}$, is bounded from the weighted Bergman space $A^p_ν$, where $0 0, $$ are established for arbitrary radial weights $ω$, $ν$ and $η$. Moreover, differences and interrelationships between the cases $A^p_ν\to L^p_ν$, $L^p_ν\to L^p_ν$ and $L^p_ν\to L^{p,\infty}_ν$ are analyzed.