How Summable are Rademacher Series?

Abstract

AbstractKhintchin inequalities show that a.e. convergent Rademacher series belong to all spaces L p([0, 1]), for finite p. In 1975 Rodin and Semenov considered the extension of this result to the setting of rearrangement invariant spaces. The space L N of functions having square exponential integrability plays a prominent role in this problem.Another way of gauging the summability of Rademacher series is considering the multiplicator space of the Rademacher series in a rearrangement invariant space X, that is, $$ \Lambda (\mathcal{R}, X) : = \left\{ {f:[0,1] \to \mathbb{R}: f \cdot \sum {\alpha _n r_n } \in X, for all \sum {\alpha _n r_n } \in X } \right\}. $$. The properties of the space Λ(R, X) are determined by its relation with some classical function spaces (as L N and L ∞([0, 1])) and by the behavior of the logarithm in the function space X.In this paper we present an overview of the topic and the results recently obtained (together with Sergey V. Astashkin, from the University of Samara, Russia, and Vladimir A. Rodin, from the State University of Voronezh, Russia.)Mathematics Subject Classification (2000)Primary 46E3546E30Secondary 47G10KeywordsRademacher functionsrearrangement invariant spaces