Radial averaging operator acting on Bergman and Lebesgue spaces

Abstract

It is shown that the radial averaging operator $$T_{\omega }(f)(z)=\frac{{\int }_{|z|}^{1}f\left(s\frac{z}{|z|}\right)\omega (s)ds}{\hat{\omega }(z)},\hat{\omega }(z)={\int }_{|z|}^1\omega (s)ds,$$ induced by a radial weight $\omega$ on the unit disc $\mathbb{D}$, is bounded from the weighted Bergman space $A_{\nu }^p$, where $0<p<\infty$ and the radial weight $\nu$ satisfies $\hat{\nu }(r)\le C\hat{\nu }\left(\frac{1+r}{2}\right)$ for all $0\le r<1$, to $L_{\nu }^p$ if and only if the self-improving condition ${\sup }_{0\le r<1}\frac{\hat{\omega }(r{)}^p}{{\int }_r^{1}s\nu (s)ds}{\int }_0^r\frac{t\nu (t)}{\hat{\omega }(t{)}^p}dt<\infty$ is satisfied. Further, two characterizations of the weak type inequality $$\eta \left(\left\{z\in \mathbb{D}:|T_{\omega }(f)(z)|\ge \lambda \right\}\right)\lesssim {\lambda }^{-p}\Vert f{\Vert }_{L_{\nu }^p}^p,\lambda >0,$$ are established for arbitrary radial weights $\omega$, $\nu$ and $\eta$. Moreover, differences and interrelationships between the cases $A_{\nu }^p\to L_{\nu }^p$, $L_{\nu }^p\to L_{\nu }^p$ and $L_{\nu }^p\to L_{\nu }^{p,\infty }$ are analyzed.