Sharp embeddings of Besov spaces involving only logarithmic smoothness

Abstract

We use Kolyada's inequality and its converse form to prove sharp embeddings of Besov spaces B 0, p,r (involving the zero classical smoothness and a logarithmic smoothness with the exponent ) into LorentzZygmund spaces. We also determine growth envelopes of spaces B 0, p,r . In distinction to the case when the classical smoothness is positive, we show that we cannot describe all embeddings in question in terms of growth envelopes.

Approximation spaces, limiting interpolation and Besov spaces

“…Embeddings of B 0,b p,r into LorentzZygmund spaces have been established by the present authors in [10] (see also the paper by Caetano, Gogatishvili and Opic [9] for the case of Besov spaces on R n and the report by Triebel [37]). Now we show embeddings into spaces Y p,r,γ introduced in Section 2.…”

Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness

“…To achieve our first goal, we use Kolyada's inequality (see [16]) and its converse form (see [3,Proposition 3.5]) to characterize the given local embedding by means of a reverse Hardy inequality restricted to the cone of non-increasing functions (see Theorem 3.1 below). Then we apply results of [9] and [10] (together with Theorem 5.3 below), to solve such an inequality completely and to obtain the desired characterization of embeddings of Besov spaces B 0,b p,r into classical Lorentz spaces Λ loc q (ω) (cf.…”

“…The paper is a direct continuation of [3], where sharp embeddings of Besov spaces B 0,b p,r into Lorentz-Karamata spaces L loc p,q;b were established and the growth envelope of the space B 0,b p,r was determined in the particular case where b(t) = ℓ β (t) andb = ℓ γ (t) with ℓ(t) := 1+| ln t|, t > 0, and γ ∈ R. Note also that our approach in [3] was more complicated. First, we have converted the given embedding to a weighted inequality, which is more involved than that of Theorem 3.1 (cf.…”

“…First, we have converted the given embedding to a weighted inequality, which is more involved than that of Theorem 3.1 (cf. [3,Proposition 3.6]). Also we have discretized the weighted inequality to find sufficient conditions for the validity of the embedding in question.…”

Compact embeddings of besov spaces involving only slowly varying smoothness

“…A lot of attention has been paid to optimal embeddings and to growth and continuity envelopes of such spaces (see, e.g., [15], [17], [22], [6], [7], [14], [2], [20], [5], [18], [19], [24,Chapt. 1], [16], [3], [4], etc.). This paper is a direct continuation of [4], where local embeddings of Besov spaces B p,r are defined by means of the modulus of continuity and they involve the zero classical smoothness and a slowly varying smoothness b.…”