An optimal symbolic calculus on Besov algebras

Abstract

In this paper we consider Besov algebras on R, that is Besov spaces B s p,q (R) for s > 1/p. For s > 1 + (1/p), p > 4/3, and q p we prove that the above algebras have a maximal symbolic calculus in the following sense: for any function f belonging locally to B s p,q (R) and such that f (0) = 0, the associated superposition operator T f (g) := f • g takes B s p,q (R) to itself.

Composition operators on Lizorkin–Triebel spaces

“…The present paper is a continuation of [7,16,17], where first characterizations as in Theorem 1(i) have been proved.…”

“…We refer to [18] and [31] also for further historical remarks. In our earlier contributions to the subject, see in particular [16], we have also dealt with Besov spaces B s p,q (R). These spaces also generalize Slobodeckij spaces but in a different way than LizorkinTriebel spaces.…”

“…see (16) for the definition of M s p,q . From the embedding F s−1 p,q → BV sp−1 , see (22), and Proposition 14, we finally derive…”

The composition in multidimensional Triebel–Lizorkin spaces

“…We note that the composition operator problem in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ B_{{p},{q}}^{s}({{\mathbb R}}^n)\cap L_\infty ({{\mathbb R}}^n) $\end{document} and in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ F_{{p},{q}}^{s}({\mathbb R}^n)\cap L_\infty ({{\mathbb R}}^n) $\end{document} is not trivial in the sense that the function f need not be linear, see e.g. 6–9. To study composition operators on intersections has a certain history: in Sobolev spaces by Adams and Frazier 1, 2, and in fractional Sobolev spaces by Brezis and Mironescu 10 and by Maz'ya and Shaposhnikova 13.…”

Superposition in homogeneous and vector valued Sobolev spaces

“…We refer to our preceding papers [10,11] and to the work of Allaoui [3] for the first steps in this direction.…”