Groupoid actions on C*-bundles and inverse semigroup actions on C*-algebras are closely related when the groupoid is r-discrete.

Normal functions and Bloch functions are respectively functions of bounded spherical expansion and bounded Euclidean expansion. In this paper we discuss the behaviour of normal functions and of Bloch functions in terms of the maximal ideal space of H∞, the Bergman projection and the Ahlfors-Shimizu characteristic.

A Radon-Nikodým theorem for Banach valued polymeasures is proved.

The authors establish the boundedness on the Herz spaces and the weak Herz spaces for a large class of rough singular integral operators and their corresponding fractional versions. Applications are given to Fefferman's rough singular integral operators, their fractional versions, their commutators with BMO() functions and Ricci-Stein oscillatory singular integral operators. Some new results are obtained.

The classic Banach Contraction Principle assumes that the self-map is a contraction. Rather than requiring that a single operator be a contraction, we weaken this hypothesis by considering a minimum involving a set of iterates of that operator. This idea is a central motif for many of the results of this paper, in which we also study how this weakended hypothesis may be applied in Caristi's theorem, and how combinatorial arguments may be used in proving fixed-point theorems.