Weighted norm inequality for a maximal operator on homogeneous space (Q2427610)

Summary: Let \(X=G/H\) be a homogeneous space, \(\tilde X=X\times[0,\infty)\), \(\mu\) a doubling measure on \(X\) induced by a Haar measure on the group \(G\), \(\beta\) a positive measure on \(\tilde X\) and \(W\) a weight on \(X\). Consider the maximal operator given by \[ \mathcal{M}f(x,r)=\sup_{s\geq r}\frac{1}{\mu(B(x,s))}\int_{B(x,s)}|f(y)|\,d\mu (y),\quad (x,r)\in\tilde X. \] In this paper, we obtain, for each \(p\), \(q\), \(1<p\geq q<\infty\), a necessary and sufficient condition for the boundedness of the maximal operator \(\mathcal{M}\) from \(L^p(X, Wd\mu)\) to \(L^q(\tilde X, d\beta)\). 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 \(\mathbb{R}^{n+1}\).