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

• Published in: Mathematische annalen Vol. 388; no. 4; pp. 4487 - 4510
• Main Authors: Ikeda, Masahiro, Ishikawa, Isao, Taniguchi, Koichi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024; Springer Nature B.V
• Subjects: Composition; Euclidean geometry; Euclidean space; Function space; Mathematics; Mathematics and Statistics; Multipliers; Operators (mathematics); Sobolev space; Topology; Primary 47B33; 30H25; Secondary 46E35
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-023-02637-3

This paper aims to characterize boundedness of composition operators on Besov spaces B p , q s 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 ≤ ∞ , 0 < q ≤ ∞ , 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.