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

TL;DR

This paper characterizes when composition operators are bounded on higher order Besov spaces in one dimension, extending known results for lower order cases and relating them to pointwise multipliers.

Contribution

It provides necessary and sufficient conditions for boundedness of composition operators on higher order Besov spaces, including non-homeomorphic maps, and relates these to pointwise multipliers.

Findings

Established conditions for boundedness of composition operators on Besov spaces.

Extended analysis to maps with bounded inverse image cardinality.

Provided similar characterizations for Sobolev spaces.

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.