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, $B_{\text{p},\text{q}}^{\mathit{w}}(\mathrm{\Omega })$ and $F_{\text{p},\text{q}}^{\mathit{w}}(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is a regular domain. We use the already known decomposition with non-smooth atoms for the spaces $B_{\text{p},\text{q}}^{\mathit{w}}(\text{rn})$ and $F_{\text{p},\text{q}}^{\mathit{w}}(\text{rn})$ in order to get the main result.