A generalized weighted Hardy-Ces\`{a}ro operator, and its commutator on weighted $L^p$ and BMO spaces

TL;DR

This paper introduces a new weighted Hardy-Cesàro operator, characterizes its boundedness on weighted $L^p$ and BMO spaces, and analyzes the boundedness of its commutators with symbols in BMO, extending previous results.

Contribution

It defines a generalized weighted Hardy-Cesàro operator, characterizes conditions for boundedness on weighted spaces, and studies the boundedness of its commutators with BMO symbols.

Findings

Characterized the weight functions for boundedness of the operator.

Derived operator norms for the Hardy-Cesàro operator.

Established necessary and sufficient conditions for commutator boundedness.

Abstract

In this paper, we introduce a new weighted Hardy-Ces\`{a}ro operator defined by $U_{\psi,s}f(x)=\int\limits_0^1 f(s(t)\cdot x) \psi(t)dt$, which is associated to the parameter curve $s(t,x)=s(t)x$. Under certain conditions on $s(t)$ and on an absolutely homogeneous weight function $\omega$, we characterize the weight function $\psi$ such that $U_{\psi,s}$ is bounded on $L^p(\omega)$, $BMO(\omega)$. The corresponding operator norms are worked out too. These results extend the ones of Jie Xiao \cite{xiao}. We also give a sufficient and a necessary condition on the weight function $\psi$, which ensure the boundedness of the commutators of operator $U_{\psi,s}$ on $L^p(\omega)$ with symbols in $BMO(\omega)$.