Intrinsic characterization and the extension operator in variable exponent function spaces on special Lipschitz domains

We study 2-microlocal Besov and Triebel-Lizorkin spaces with variable exponents on special Lipschitz domains. These spaces are as usual defined by restriction of the corresponding spaces on ${\mathbb{R}}^n$. In this paper we give two intrinsic characterizations of these spaces using local means and the Peetre maximal operator. Further we construct a linear and bounded extension operator following the approach done by Rychkov, which at the end also turns out to be universal.