Weighted Norm Inequality for a Maximal Operator on Homogeneous Space

Abstract

Let $X=G/H$ be a homogeneous space, $X=X\times [0,\infty)$, $\mu$ a doubling measure on $X$ induced by a Haar measure on the group $G$, $\beta$ a positive measure on $X$ and $W$ a weight on $X$. Consider the maximal operator given by

In this paper, we obtain, for each $p, q, 1<p\leq q<\infty$, a necessary and sufficient condition for the boundedness of the maximal operator ${\cal M}$ from $L^p(X, Wd\mu)$ to $L^q( X, d\beta)$. As an application, we obtain a necessary and sufficient condition for the boundedness of the Poisson integral of functions defined on the unit sphere $S^n$ of the Euclidian space $\mathbb{R}^{n+1}$, from $L^p(S^n,Wd\sigma)$ to $L^q(\mathbb{B}, d\nu)$, where $\sigma$ is the Lebesgue measure on $S^n$, $W$ is a weight on $S^n$ and $\nu$ is a positive measure on the unit ball $\mathbb{B}$ of $\mathbb{R}^{n+1}$.