Maximal operators and strong differentiability of the integral (Q1057441)

Let H be a collection of open bounded sets in \({\mathbb{R}}^ n\) such that \(\cup_{R\in H}R={\mathbb{R}}^ n,\) and let \(\phi_ R\geq 0\) be Borel measurable functions associated with each \(R\in H\), \(\sup p \phi_ R\subseteq R.\) Let \(\nu\geq 0\) be a Borel measure on \({\mathbb{R}}^ n\) and define \(M^*f(x)=\sup_{x\in R\in H}\int f \phi_ Rd\nu.\) In this note the author announces a number of rearrangement inequalities for these operators. In particular, an application is given to the study of a weighted version of the strong maximal function.