Superposition operators and functions of bounded p-variation II

We continue the study of the superposition operator Tf:g↦f∘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 BVp1(I) to itself. Then we characterize the functions f for which Tf is continuous, uniformly continuous, and differentiable, as a mapping of BVp1(I) to itself, respectively. By exploiting the Peetre's Imbedding Theorem and the Fubini property, we derive partial results on continuity of Tf in Besov spaces Bp,qs(Rn), for a smoothness parameter s satisfying 0