Weighted norm inequality for the Poisson integral on the sphere (Q701335)

The authors consider a homogeneous space \(X= G/H\) where \(G\) is a locally compact Hausdorff topological group and \(H\) is a compact subgroup of \(G\) which is provided with a quasi-distance \(d\) and with a measure \(\mu\) induced on \(X\) by a Haar measure on \(G\). Denote \(\widetilde X= X\times [0,\infty)\) and \(\widetilde B= B(x,r)\times [0,r]\) with any ball \(B(x,r)\) in \(X\). Let \(\beta\) be a positive Borel measure on \(\widetilde X\) and \(W\) be a weight on \(X\). Define the maximal operator \({\mathcal M}\) by \[ {\mathcal M}f(x,r)= \sup_{s\geq r} {1\over \mu(B(x, s))} \int_{B(x,s)}|f(y)|d\mu(y). \] The authors prove the following theorem: Theorem 3.1. Let \(G\) be a compact or an Abelian group, let \(1< p< \infty\) and let \(W\) be a weight on \(X\) such that \(W^{1-p'}\in A_\infty(X)\). Then the following conditions are equivalent: (i) There exists a constant \(C>0\), such that, for all \(f\in L^p(W)\) \[ \int_{\widetilde X} \{{\mathcal M}f(x,r)\}^p d\beta(x,r)\leq C \int_X|f(x)|^p W(x) d\mu(x). \] (ii) There exists a constant \(C> 0\), such that for all balls \(B(z,t)\), \[ \int_{\widetilde B}\{{\mathcal M}(W^{1-p'}\chi_B)(x,r)\}^p d\beta(x, r)\leq C\int_B W^{1- p'}(x) d\mu(x)< \infty. \] The above result is an extension of the known result on \(\mathbb{R}^n\). As an application, the authors obtain, for each \(p\), \(1< p< \infty\), a necessary and sufficient condition for the Poisson integral of functions defined on the unit sphere \(S^n\), to be bounded from a weighted space \(L^p(S^n, Wd(\sigma))\) into a space \(L^p(\mathbb{B},\nu)\), where \(\nu\) is a positive measure on the unit ball \(\mathbb{B}\) of \(\mathbb{R}^{n+1}\).