On Dilation Operators in Besov Spaces

Abstract

We consider dilation operators

On sharp embeddings of Besov and Triebel-Lizorkin spaces in the subcritical case

“…These authors worked with spaces defined by differences and their results differ from ours in logarithmic factors. This shows indirectly that the Fourier-analytical definition and the classical definition of Besov spaces do not coincide for s = 0, an effect observed in detail recently by Schneider [9].…”

Embeddings of function spaces: a criterion in terms of differences

“…Thus the result for ' t in (20) together with (21) and (22) completes the argument for arbitrary ' 2 D(R n ).…”

“…Then B s p,q ðR n Þ consists of all f 2 L p (R n ) such that (6) is satisfied; this is also true for s ¼ p , but here the situation is more tricky and was settled only recently in [21,22]. However, when s 4 p , the outcome is optimal in the following sense.…”

On Besov spaces of logarithmic smoothness and Lipschitz spaces

“…If we take classical smoothness s and additional logarithmic smoothness with exponent b, the first way leads to spaces B s,b p,q and the second to spaces B s,b p,q (precise definitions are given in Section 2). If 1 ≤ p ≤ ∞ and s > 0, it turns out that B s,b p,q = B s,b p,q with equivalence of norms (see [30,Theorem 2.5] and [39, 2.5.12]; but if 0 < p < 1 and 0 < q ≤ 1 then as it is shown in [37,Corollary 3.10]). However, the relation between these two kinds of spaces when s = 0 has not been described yet.…”