On Besov spaces of logarithmic smoothness and Lipschitz spaces

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

“…We are mainly interested in the case s = 0. Spaces B 0,b p,q also have logarithmic smoothness but they are different from B 0,b p,q although they are closely related (see [11,Theorem 3.3] and [12,Theorem 3.2]). The characterization of B 0,b p,q in terms of Fourier-analytical decompositions has not been studied yet, even for the classical space B 0 p,q .…”

“…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.…”

“…According to [11,Theorem 3.1] one can replace ω(f, t) p in (1.16) by ω k (f, t) p , k ∈ N, which means that B 0,b p,q consists of all f ∈ L p such that…”

“…does not characterize B 0,b p,q but the Fourier-analytically defined space B 0,b p,q also with logarithmic smoothness of exponent b but which is different from B 0,b p,q (see [11,Theorem 3.3]). We show that B 0,b p,q consists of all f ∈ L p such that…”

Kolmogorov Widths for Analogs of the Nikol’skii–Besov Classes with Logarithmic Smoothness

“…Note that, for the classes LG r , which can be identified with the classes H 0,r 1 , the exact-order estimates for the Kolmogorov widths and entropy numbers were established in [1]. The classes defined by relations (1)- (3) were also studied in [2,3] from the viewpoint of finding the order estimates for some approximating characteristics of these classes and in [4] from the viewpoint of embeddings in some spaces of smooth functions.…”