JournalszaaVol. 27 , No. 1DOI 10.4171/zaa/1344

Weighted Norm Inequality for a Maximal Operator on Homogeneous Space

  • Iara A. A. Fernandes

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

    Brazil
Weighted Norm Inequality for a Maximal Operator on Homogeneous Space cover

Abstract

Let X=G/HX=G/H be a homogeneous space, X=X×[0,)X=X\times [0,\infty), μ\mu a doubling measure on XX induced by a Haar measure on the group GG, β\beta a positive measure on XX and WW a weight on XX. Consider the maximal operator given by

Mf(x,r)=supsr1μ(B(x,s))B(x,s)f(y)dμ(y),(x,r)X.{\cal M} f(x,r)=\sup _{s\geq r} \frac{1}{\mu (B(x,s))}\int_{B(x,s)} |f(y)|\,d\mu (y), \quad (x,r)\in X.

In this paper, we obtain, for each p,q,1<pq<p, q, 1<p\leq q<\infty, a necessary and sufficient condition for the boundedness of the maximal operator M{\cal M} from Lp(X,Wdμ)L^p(X, Wd\mu) to Lq(X,dβ)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 SnS^n of the Euclidian space Rn+1\mathbb{R}^{n+1}, from Lp(Sn,Wdσ)L^p(S^n,Wd\sigma) to Lq(B,dν)L^q(\mathbb{B}, d\nu), where σ\sigma is the Lebesgue measure on SnS^n, WW is a weight on SnS^n and ν\nu is a positive measure on the unit ball B\mathbb{B} of Rn+1\mathbb{R}^{n+1}.