Intrinsic atomic characterization of 2-microlocal spaces with variable exponents on domains

We provide an intrinsic atomic characterization for 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability on domains, $B_{\p,\q}^{\bm{w}}(\Omega)$ and $F_{\p,\q}^{\bm{w}}(\Omega)$, where $\Omega$ is a regular domain. We use the already known decomposition with non-smooth atoms for the spaces $B_{\p,\q}^{\bm{w}}(\rn)$ and $F_{\p,\q}^{\bm{w}}(\rn)$ in order to get the main result.