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On the relationship between two kinds of Besov spaces with smoothness near zero and some other applications of limiting interpolation. (English) Zbl 1365.46027
Summary: 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 $$\mathbf B^{0,-1/q}_{p,q}(\mathbb {T})$$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46M35 Abstract interpolation of topological vector spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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