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

Abstract

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