Superposition operators and functions of bounded p-variation II

We continue the study of the superposition operator $T_{f}: g\mapsto f\circ g$, on the space $BV^{1}_{p}(I)$ 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 $T_f$ takes $BV^{1}_{p}(I)$ to itself. Then we characterize the functions $f$ for which $T_f$ is continuous, uniformly continuous, and differentiable, as a mapping of $BV^{1}_{p}(I)$ to 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 $B^{s}_{p,q}({\mathbb{R}}^n)$, for a smoothness parameter $s$ satisfying $0<s\leq 1+(1/p)$.