Abstract
It is shown that the radial averaging operator
T_{\omega}(f)(z)=\frac{\int_{|z|}^{1}f\bigl{(}s\frac{z}{|z|}\bigr{)}\omega(s)%
\,ds}{\widehat{\omega}(z)},\quad\widehat{\omega}(z)=\int_{|z|}^{1}\omega(s)\,ds,
induced by a radial weight ω on the unit disc
{\mathbb{D}}
, is bounded from the weighted Bergman space
{A^{p}_{\nu}}
, where
{0<p<\infty}
and the radial weight ν satisfies
{\widehat{\nu}(r)\leq C\widehat{\nu}(\frac{1+r}{2})}
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(\{z\in\mathbb{D}:|T_{\omega}(f)(z)|\geq\lambda\})\lesssim\lambda^{-p}\|f%
\|_{L^{p}_{\nu}}^{p},\quad\lambda>0,
are established for arbitrary radial weights ω, ν and η. 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.
An Extension of Nikishin’s Factorization Theorem
