Locally Lipschitz composition operators in the space ΦBV[a,b]ΦBV[a,b]

We give a necessary and sufficient condition on a function f:R→Rf:R→R such that the composition operator (Nemytskii operator) FF defined by Fh=f∘hFh=f∘h acts in the space ΦBV[a,b]ΦBV[a,b] and satisfies a local Lipschitz condition. This complements recent results on the spaces BVφW[a,b], HBV[a,b]HBV[a,b] and RVφ[a,b]RVφ[a,b]. In the proof of the main theorem, we use a Helly’s type selection principle.