Oscillations and differences in Triebel-Lizorkin-Morrey spaces

In this paper we are concerned with Triebel-Lizorkin-Morrey spaces ${\mathcal{E}}_{u,p,q}^s(\mathrm{\Omega })$ of positive smoothness $s$ defined on (special or bounded) Lipschitz domains $\mathrm{\Omega }\subset {\mathbb{R}}^d$ as well as on ${\mathbb{R}}^d$. For those spaces we prove new equivalent characterizations in terms of local oscillations which hold as long as some standard conditions on the parameters are fulfilled. As a byproduct, we also obtain novel characterizations of ${\mathcal{E}}_{u,p,q}^s(\mathrm{\Omega })$ using differences of higher order. Special cases include standard Triebel-Lizorkin spaces $F_{p,q}^s(\mathrm{\Omega })$ and hence classical $L_p$-Sobolev spaces $H_p^s(\mathrm{\Omega })$. Key words: Triebel-Lizorkin-Morrey space, Morrey space, Lipschitz domain, oscillations, higher order differences