  In this paper we provide non-smooth atomic decompositions of 2-microlocal
Besov-type and Triebel-Lizorkin-type spaces with variable exponents
$B^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and
$F^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$. Of
big importance in general, and an essential tool here, are the
characterizations of the spaces via maximal functions and local means, that we
also present. These spaces were recently introduced by Wu at al. and cover not
only 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 also
the 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 pointwise
multipliers assertion for this scale.
