Composition operators on Besov algebras

Abstract

We study the composition operator T f (g) := f • g on Besov spaces B s p,q (R). In the case 1 < p < +∞, p ≤ q ≤ +∞ and s > 1 + (1/p) we will prove that the operator T f takes B s p,q (R) into itself if and only if f (0) = 0 and f belongs locally to B s p,q (R). • If 1 < p < +∞, s > n/p and f ∈ C ∞ (R), then T f takes the Bessel potential spaces H s p (R n) (resp. B s p,q (R n)) into itself, see e.g. Meyer [12], (resp. Peetre [13]). • Dahlberg [9] proved that for 1 ≤ p ≤ +∞ and 1 + (1/p) < m < n/p (m integer), if T f maps the Sobolev spaces W m p (R n) into itself, then f is a linear function. Also, after the work of Marcus and Mizel [11] on W 1 p (R n), Bourdaud [2] proved that for either m ≥ 2 and mp > n, or m = n ≥ 2 and p = 1, the operator T f maps W m p (R n) into itself if and only if f ∈ W m, oc p (R). • The result of Dahlberg has been extended to B s p,q (R n). More precisely, for 1 ≤ p, q ≤ +∞ and either 1 + (1/p) < s < n/p, or 1 + (1/p) = s < n/p and 1 < q ≤ +∞, if the operator T f acts on B s p,q (R n) then f (t) = ct for some real c, see e.g. [15] and [16]. The characterization of those functions f such that T f acts on B s p,q (R n) requires to investigate the necessary conditions. In this sense, we recall the following two results:

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

“…Thus the standard facts from the theory of Besov spaces (duality, isomorphisms, etc) cannot a priori be applied to (B, • B ). The algebra B has appeared, for example, in [76], and similar algebras were considered in [59]. Now we briefly discuss the homogeneous class E dyad := B 0+ 1,∞ (R) with the corresponding norm…”

Hit and Lead Profiling: Identification and Optimization of Drug‐like Molecules. Edited by B. Faller and L. Urban.

“…This result is a variant of the Nikol'skij representation method, see 10, Proposition 2.3.2(1), p. 59], 12, 4, Proposition 4], 7, Proposition 2].…”

Composition operators acting on Besov spaces on the real line

“…In our preceding papers [13][14][15]20], we always used, as the first step of the proof of Theorem 2, some arguments to simplify the situation. We will do this here as well.…”