Generalized Hardy-Cesàro operators between weighted spaces

Abstract

We characterize those non-negative, measurable functions $\psi$ on $[0,1]$ and positive, continuous functions ${\omega }_1$ and ${\omega }_2$ on ${\mathbb{R}}^{+}$ for which the generalized Hardy-Cesàro operator $$(U_{\psi }f)(x)={\int }_0^{1}f(tx)\psi (t)dt$$ defines a bounded operator $U_{\psi }:L^1({\omega }_1)\to L^1({\omega }_2)$. Furthermore, we extend $U_{\psi }$ to a bounded operator on $M({\omega }_1)$ with range in $L^1({\omega }_2)\oplus \mathbb{C}{\delta }_0$. Finally, we show that the zero operator is the only weakly compact generalized Hardy-Cesàro operator from $L^1({\omega }_1)$ to $L^1({\omega }_2)$.