Composition operators acting on Besov spaces on the real line

Abstract

We study the composition operator T f (g) := f • g on Besov spaces B s p,q (R). In case 1 < p < +∞, 0 < q ≤ +∞ and s > 1 + (1/ p), we will prove that the operator T f maps B s p,q (R) to itself if, and only if, f (0) = 0 and f belongs locally to B s p,q (R). For the case p = q, i.e., in case of Slobodeckij spaces, we can extend our results from the real line to R n . KeywordsHomogeneous and inhomogeneous Besov spaces on the real line • Slobodeckij spaces on R n • Functions of bounded p-variation • Composition operators • Optimal inequalities Mathematics Subject Classification (2000) 46E35 • 47H30 1 IntroductionThe present paper is a continuation of our earlier investigations, see [13][14][15] as well as [20], on composition operators, i.e., mappings

Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

“…However, we prove in Theorem 5.3 that f acts on the periodic Besov space B s pq (T), 1 < p < ∞, 1 < q ≤ ∞ and s > 1+ 1 p , if, and only if, f ∈ B s pq,loc (R), where T denotes the one-dimensional torus (the requirement that f (0) = 0 appearing in [3] is not necessary due to the periodicity, else the results are comparable). Our proof relies on the composition theorem for non-periodic Besov spaces and what can be called a localizing property of periodic Besov spaces (see Lemma 5.2): Smooth but compactly supported extensions of periodic functions from a n-dimensional torus T n to the whole space R n will be controlled by the latter periodic function norms.…”

“…The key for such a commutator estimate in our case are the mapping properties of the nonlinearity n over the energy spaces in consideration, which is guaranteed by a composition theorem [3] when the initial data are in Sobolev spaces on the line. Namely, a composition operator T f is said to act on a function space X if…”

“…The composition theorem in [3] holds that a function f acts on Besov spaces B s pq (R), 1 < p < ∞, 0 < q ≤ ∞, s > 1+1/ p, if, and only if, f ∈ B s pq,loc (R) and f (0) = 0. That composition theorem, when applied to a non-smooth nonlinearity n, guarantees a contraction mapping in the desired solution spaces.…”

“…In the periodic setting, a composition theorem like the one in [3] is not available in the literature. However, we prove in Theorem 5.3 that f acts on the periodic Besov space B s pq (T), 1 < p < ∞, 1 < q ≤ ∞ and s > 1+ 1 p , if, and only if, f ∈ B s pq,loc (R), where T denotes the one-dimensional torus (the requirement that f (0) = 0 appearing in [3] is not necessary due to the periodicity, else the results are comparable).…”

“…Our proof relies on the composition theorem for non-periodic Besov spaces and what can be called a localizing property of periodic Besov spaces (see Lemma 5.2): Smooth but compactly supported extensions of periodic functions from a n-dimensional torus T n to the whole space R n will be controlled by the latter periodic function norms. A feature of our proof is that we work directly with the difference-derivative characterization of Besov spaces, not the Littlewood-Paley decompositions (used for example in [3]). In this case, the main difficulty arises at integer regularity s ∈ N, where the estimates are particularly cumbersome.…”