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In this article we study atomic and molecular decompositions in$2$-microlocal Besov and Triebel--Lizorkin spaces with variable integrability.We show that, in most cases, the convergence implied in such decompositionsholds not only in the distributions sense, but also in the function spacesthemselves. As an application, we give a simple proof for the denseness of theSchwartz class in such spaces. Some other properties, like Sobolev embeddings,are also obtained via atomic representations.

Atomic and molecular decompositions in variable exponent 2-microlocal spaces and applications