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

Abstract

We characterize local embeddings of Besov spaces B 0,b p,r involving only a slowly varying smoothness b into classical Lorentz spaces. These results are applied to establish sharp local embeddings of the Besov spaces in question into Lorentz-Karamata spaces. As a consequence of these results, we are able to determine growth envelopes of spaces B 0,b p,r and to show that we cannot describe all local embeddings of Besov spaces B 0,b p,r into Lorentz-Karamata spaces in terms of growth envelopes.

Optimal local embeddings of Besov spaces involving only slowly varying smoothness

“…As in [CGO11b], also in our paper Besov spaces are defined by means of the modulus of continuity. Note that some authors use the Fourier-analytical approach to define Besov spaces with the zero classical smoothness and involving the logarithmic smoothnees b, where (1.11) b(t) := (1 + | ln t|) α , t ∈ (0, +∞), with a convenient α ∈ IR.…”

“…In [CGO11b] the authors have characterized (with easily verifiable conditions) local embeddings of Besov spaces B 0,b p,r involving the zero classical smoothness and involving only a slowly varying smoothness b into classical Lorentz spaces. These results have been then applied to establish sharp local embeddings of Besov spaces in question into Lorentz-Karamata spaces and to determinate the growth envepoles of spaces B 0,b p,r .…”

Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-groups

“…Among them, logarithmically perturbed Besov spaces are receiving a growing interest in recent times as can be seen in the papers by Caetano, Gogatishvili and Opic [6,7], Besov [4] or Cobos and Domínguez [9][10][11]. These spaces have classical smoothness zero and logarithmic smoothness with exponent b.…”

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.…”

“…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. 1 In particular, the following two theorems are proved there.…”