A maximal operator and a covering lemma on non-compact symmetric spaces
Let \(X\) be a symmetric space of noncompact type. The boundedness of the noncentred Hardy-Littlewood maximal operator \[ {\mathcal M}_{\mathcal F}f(z)= \sup_{z\in B\in{\mathcal F}} {1\over|B|} \int_B|f(x)|dx \] is studied. Here \(|B|\) denotes the measure of a ball \(B\) and \({\mathcal F}\) is the family of all balls in \(X\). The balls in a symmetric space of noncompact type do not have a doubling property, therefore the properties of the above maximal operators differ from the properties of a centered maximal operator, with supremum taken over balls centered at \(z\). The last operator is known to be of weak \((1,1)\) type and strong \((p,p)\) type, \(p>1\). The main result of the paper asserts that \({\mathcal M}_{\mathcal F}\) is a bounded operator in \(L^p(X)\) for \(p\in (2,\infty]\). Moreover, it is proved that \({\mathcal M}_{\mathcal F}\) is not bounded from the Lorentz space \(L^{2,\alpha}(X)\) to \(L^{2,\infty}(X)\) if \(\alpha> 1\). So the range of \(p\) for which \({\mathcal M}_{\mathcal F}\) is bounded is optimal. The proof is based on the notion of \(V_p\) covering property introduced by A. Cordoba and R. Fefferman.
scientific article; zbMATH DE number 1489545