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In this paper we provide non-smooth atomic decompositions of 2-microlocalBesov-type and Triebel-Lizorkin-type spaces with variable exponents$B^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and$F^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$. Ofbig importance in general, and an essential tool here, are thecharacterizations of the spaces via maximal functions and local means, that wealso present. These spaces were recently introduced by Wu at al. and cover notonly variable 2-microlocal Besov and Triebel-Lizorkin spaces$B^{\mathrm{\boldsymbol{\omega}}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and$F^{\mathrm{\boldsymbol{\omega}}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$, but alsothe more classical smoothness Morrey spaces $B^{s, \tau}_{p,q}(\mathbb{R}^n)$and $F^{s,\tau}_{p,q}(\mathbb{R}^n)$. Afterwards, we state a pointwisemultipliers assertion for this scale.

Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces