Generalized Hardy-Ces\`aro operators between weighted spaces

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-Ces`aro operator $$(U_{\psi }f)(x)={\int }_0^{1}f(tx)\psi (t)dt$$ defines a bounded operator $U_{\psi}:L^1(\omega_1)\to L^1(\omega_2)$\mathrm{.}Furthermore,weextend$U_{\psi}$toaboundedoperatoron$M(\omega_1)$withrangein$L^1(\omega_2)\oplus\mathbb C\delta_0$. Finally, we show that the zero operator is the only weakly compact generalized Hardy-Ces`aro operator from $L^1({\omega }_1)$ to $L^1({\omega }_2)$.