Maximal operators and differentiation associated to collections of shifted convex bodies

Maximal operators and differentiation of integrals associated to collections of shifted balls in \(\mathbb{R}^n\) (i.e., balls that may not contain the origin) have been studied by various authors. One of the motivations has been the intimate connection of these concepts with the boundary behaviour of Poisson integrals along regions more general than cones. Generalizing the corresponding results of Nagel and Stein, and Hagelstein and Parissis (established for the case of collections of balls) we give characterizations of the two classes of monotone collections \(\Omega\) of shifted convex bodies in \(\mathbb{R}^n\) that are defined by the following properties respectively: 1) the maximal operator associated to \(\Omega\) (i.e., to the means \((1/|B|)\int_{B+x}|f|\) \((B\in \Omega)\)) satisfies the weak type \((1,1)\) inequality; 2) the means over the sets \(B+x\) \((B\in\Omega)\) are a.e. convergent for the characteristic function of an arbitrary measurable subset of \(\mathbb{R}^n\).