  In this paper we present a comprehensive treatment of function spaces with
logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the
following results: Sharp embeddings between the Besov spaces defined by
differences and by Fourier-analytical decompositions as well as between Besov
and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov
norms in terms of different K-functionals. For instance, we derive
characterizations via ball averages, approximation methods, heat kernels, and
Bianchini-type norms; Sharp estimates for Besov norms of derivatives and
potential operators (Riesz and Bessel potentials) in terms of norms of
functions themselves. We also obtain quantitative estimates of regularity
properties of the fractional Laplacian. The key tools behind our results are
limiting interpolation techniques and new characterizations of Besov and
Sobolev norms in terms of the behavior of the Fourier transforms for functions
such that their Fourier transforms are of monotone type or lacunary series.
