Boundedness of composition operators on higher order Besov spaces in one dimension

Abstract

This paper aims to characterize boundedness of composition operators on Besov spaces $$B^s_{p,q}$$ of higher order derivatives $$s>1+1/p$$ on the one-dimensional Euclidean space. In contrast to the lower order case $$0<s<1$$ , there were a few results on the boundedness of composition operators for $$s>1$$ . We prove a relation between the composition operators and pointwise multipliers of Besov spaces, and effectively use the characterizations of the pointwise multipliers. As a result, we obtain necessary and sufficient conditions for the boundedness of composition operators for general p, q, and s such that $$1<p\le \infty $$ , $$0<q\le \infty $$ , and $$s>1+1/p$$ . In this paper, we treat, as a map that induces the composition operator, not only a homeomorphism on the real line but also a continuous map whose number of elements of inverse images at any one point is bounded above. We also show a similar characterization of the boundedness of composition operators on Sobolev spaces.