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In this article we introduce Besov-type spaces with variable smoothness and integrability, which unify and generalize the Besov-type spaces with fixed exponents. Under natural regularity assumptions on the exponent functions, we show that our spaces are well-defined, i.e., independent of the choice of basis functions and we establish some properties of these function spaces. Moreover the Sobolev embeddings for these function spaces are obtained.

The aim of this paper is to study properties of Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the ϕ-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces.

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