Radial averaging operator acting on Bergman and Lebesgue spaces

It is shown that the radial averaging operator $$T_\omega(f)(z)=\frac{\int_{|z|}^1f\left(s\frac{z}{|z|}\right)\omega(s)\,ds}{\widehat{\omega}(z)},\quad \widehat{\omega}(z)=\int_{|z|}^1\omega(s)\,ds,$$ induced by a radial weight $\omega$ on the unit disc $\mathbb{D}$, is bounded from the weighted Bergman space $A^p_\nu$, where $0<p<\infty$ and the radial weight $\nu$ satisfies $\widehat{\nu}(r)\leq C\widehat{\nu}\left(\frac{1+r}{2}\right)$ for all $0\leq r<1$, to $L^p_\nu$ if and only if the self-improving condition $\sup_{0\leq r<1}\frac{\widehat{\omega}(r)^p}{\int_{r}^1 s\nu(s)\,ds}\int_0^r\frac{t\nu(t)}{\widehat{\omega}(t)^p}\,dt<\infty$ is satisfied. Further, two characterizations of the weak type inequality $$\eta \left(\left\{ z\in\mathbb{D} : |T_\omega(f)(z)|\geq\lambda\right\}\right)\lesssim\lambda^{-p} \| f\|_{L^p_\nu}^p,\quad \lambda>0,$$ are established for arbitrary radial weights $\omega$, $\nu$ and $\eta$. Moreover, differences and interrelationships between the cases $A^p_\nu\to L^p_\nu$, $L^p_\nu\to L^p_\nu$ and $L^p_\nu\to L^{p,\infty}_\nu$ are analyzed.