Generalized Hardy-Ces\`aro operators between weighted spaces

TL;DR

This paper characterizes when the generalized Hardy-Ces extbackslash ar extbackslash o operator is bounded between weighted $L^1$ spaces, extends it to a measure space, and shows that only the zero operator is weakly compact.

Contribution

It provides necessary and sufficient conditions for boundedness of the generalized Hardy-Ces extbackslash ar extbackslash o operator between weighted spaces and explores its compactness properties.

Findings

Characterization of boundedness conditions for $U_{\psi}$

Extension of $U_{\psi}$ to measure spaces

Only the zero operator is weakly compact

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-Ces\`aro operator $$(U_{\psi}f)(x)=\int_0^1 f(tx)\psi(t)\,dt$$ defines a bounded operator $U_{\psi}:L^1(\omega_1)\to L^1(\omega_2)$. 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-Ces\`aro operator from $L^1(\omega_1)$ to $L^1(\omega_2)$.