# Weighted Norm Inequality for a Maximal Operator on Homogeneous Space

### Iara A. A. Fernandes

Universidade São Francisco, Itatiba, Brazil### Sergio A. Tozoni

Brazil

## Abstract

Let $X=G/H$ be a homogeneous space, $X=X×[0,∞)$, $μ$ a doubling measure on $X$ induced by a Haar measure on the group $G$, $β$ a positive measure on $X$ and $W$ a weight on $X$. Consider the maximal operator given by

In this paper, we obtain, for each $p,q,1<p≤q<∞$, a necessary and sufficient condition for the boundedness of the maximal operator $M$ from $L_{p}(X,Wdμ)$ to $L_{q}(X,dβ)$. 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 $R_{n+1}$, from $L_{p}(S_{n},Wdσ)$ to $L_{q}(B,dν)$, where $σ$ is the Lebesgue measure on $S_{n}$, $W$ is a weight on $S_{n}$ and $ν$ is a positive measure on the unit ball $B$ of $R_{n+1}$.

## Cite this article

Iara A. A. Fernandes, Sergio A. Tozoni, Weighted Norm Inequality for a Maximal Operator on Homogeneous Space. Z. Anal. Anwend. 27 (2008), no. 1, pp. 67–78

DOI 10.4171/ZAA/1344