Abstract. The concept of local growth envelope of a quasi-normed function space is applied to the spaces of Besov and Triebel-Lizorkin type of generalized smoothness (s, Ψ) in the critical case s = n/p , where s stands for the main smoothness, Ψ is a perturbation and p stands for integrability. The expression obtained for the behaviour of the local growth envelope functions (which, as expected, depends on Ψ ) shows the ability to be generalized to a form unifying both critical ( s = n/p ) and subcritical ( s < n/p ) cases. (2000): 46E35. Mathematics subject classification

Sharp estimates for the approximation numbers of embeddings between the function spaces B s pq and F s pq on domains are given in a case not thoroughly studied by Edmunds and Triebel. Corresponding sharp estimates are also obtained for the counterparts of that case in the weighted function space setting. Academic Press

Abstract. 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 decompositions holds not only in the distributions sense, but also in the function spaces themselves. As an application, we give a simple proof for the denseness of the Schwartz class in such spaces. Some other properties, like Sobolev embeddings, are also obtained via atomic representations.

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Abstract. In this paper we study various key properties for 2-microlocal Besov and Triebel-Lizorkin spaces with all exponents variable, including the lifting property, embeddings and Fourier multipliers. We also clarify and improve some statements recently published.