Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations

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 Besov defined by differences and Fourier-analytical decompositions as well Sobolev/Triebel-Lizorkin spaces; Various new characterizations for norms in terms different K-functionals. For instance, derive via ball averages, approximation methods, heat kernels, Bianchini-type norms; estimates derivatives potential operators (Riesz Bessel potentials) functions themselves. also obtain quantitative regularity properties fractional Laplacian. The key tools behind our results are limiting interpolation techniques Sobolev behavior Fourier transforms such that their monotone type or lacunary series.