A homogeneity property for Besov spaces

Abstract

A homogeneity property for some Besov spacesBp,qsis proved. An analogous property for someFp,qsspaces is already known.

Non-smooth atomic decompositions of anisotropic function spaces and some applications

“…As we will see below, the Fubini property will play a central role in the proof of the homogeneity property for anisotropic Besov spaces B s,α p (R n ). The following proposition is a simple consequence of recent results on the homogeneity property in isotropic function spaces on domains due to A. Caetano et al [1]. Proposition 3.3.…”

“…Here ∆ m t,r f = ∆ m h f with h = te r , t ∈ R, denote the iterated differences according to (1) in the direction of the rth coordinate and e r stands for the corresponding unit vector in R n . Once again putting s 1 = · · · = s n = s > 0, we recover the classical Besov spaces as presented for instance in [10, Section 1.2.5].…”

“…For a complete treatment of the homogeneity property for isotropic Besov and Triebel-Lizorkin spaces on domains, the reader may consult [1]. The next result describes the homogeneity property in special anisotropic Besov spaces, when p = q.…”

Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces

“…66] and for Besov spaces to [8]). So, considering A ∈ {B, F }, for all 0 < p, q ≤ ∞ (with p < ∞ for the F -spaces) and s > n(1/min(1, p) − 1) (if A = B) or s > n(1/min(1, p, q) − 1) (if A = F ), In this paper we prove an adapted homogeneity property for Besov spaces of generalised smoothness.…”

“…Now one is in the same situation as in [8]. In that case the integration region was R n instead of {u : |u| ≤ 2 −εJ } but this is immaterial.…”