Besov spaces with variable smoothness and integrability

In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove independence of the choice of basis functions, as well as several other basic properties. We also give Sobolev-type embeddings, and show that our scale contains variable order Hölder–Zygmund spaces as special cases. We provide an alternative characterization of the Besov space using approximations by analytic functions.