On superposition operators in spaces BVφ(0,1)

For a function f:[0,1]×R→Rf:[0,1]×R→R the superposition operator Sf:R[0,1]→R[0,1]Sf:R[0,1]→R[0,1] is defined by the formula Sf(φ)(t)=f(t,φ(t))Sf(φ)(t)=f(t,φ(t)). We study properties of operators SfSf in Banach spaces BVφ(0,1)BVφ(0,1) of all real functions of bounded φ  -variation on [0,1][0,1] for convex functions φ  . We show that if an operator SfSf maps the space BVφ(0,1)BVφ(0,1) into itself, then (1) it maps bounded subsets of BVφ(0,1)BVφ(0,1) into bounded sets if additionally f   is locally bounded, (2) f=fcr+fdrf=fcr+fdr where the operator SfcrSfcr maps the space D(0,1)∩BVφ(0,1)D(0,1)∩BVφ(0,1) of all right-continuous functions in BVφ(0,1)BVφ(0,1) into itself and the operator SfdrSfdr maps the space BVφ(0,1)BVφ(0,1) into its subset consisting of functions with countable support. Moreover we show that if an operator SfSf maps the space D(0,1)∩BVφ(0,1)D(0,1)∩BVφ(0,1) into itself, then f is locally Lipschitz in the second variable uniformly with respect to the first variable.