On the Relationship Between Two Kinds of Besov Spaces with Smoothness Near Zero and Some Other Applications of Limiting Interpolation

Using limiting interpolation techniques we study the relationship between Besov spaces B-p,q(0,-1/q) (T) with zero classical smoothness and logarithmic smoothness defined by means of differences with similar spaces defined by means of the Fourier transform. Among other things, we prove that B-2,2(0,-1/2) = B-2,2(0,0,-1/2). We also derive several results on periodic spaces B-p,q(0,-1/q) (T), including embeddings in generalized Lorentz-Zygmund spaces and the distribution of Fourier coefficients of functions of B-p,q(0,-1/q) (T).