Superposition operators and functions of bounded p-variation II

We continue the study of the superposition operator T-f : g -> f o g, on the space BVp1 (I) of primitives of real-valued functions of bounded p-variation on an interval I. We give first a characterization of the functions f such that Tf takes B V 1 (1) to itself. Then we characterize the functions f P for which Tf is continuous, uniformly continuous, and differentiable, as a mapping of B V 1 (1) to P itself, respectively. By exploiting the Peetre's Imbedding Theorem and the Fubini property, we derive partial results on continuity of T-f in Besov spaces B-p,q(s) (R-n), for a smoothness parameters satisfying 0 < s <= 1 + (1/p). (c) 2005 Elsevier Ltd. All rights reserved.