Modulation Spaces with Variable Smoothness and Integrability

This paper introduces modulation spaces with variable smoothness and integrability, defined via frequency-uniform decomposition operators and mixed Lebesgue-sequence spaces. Since the conventional dyadic decomposition is replaced by a uniform one, a new theoretical foundation is required. Therefore, we first introduce a new sequence of functions and establish some important results related to these functions, which are fundamental to our analysis. We then demonstrate that the definition of these modulation spaces is independent of the choice of basis functions. Furthermore, we establish several embedding theorems and prove the completeness properties of these spaces.