Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability

Abstract

We introduce 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability and give characterizations by decompositions in atoms, molecules and wavelets. These spaces cover the usual Besov and Triebel-Lizorkin spaces as well as spaces of variable smoothness and integrability. We emphasize that the spaces $F^{s(cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ which were defined recently in [12] are included in this approach.