Oscillations and differences in Triebel–Lizorkin–Morrey spaces

In this paper we are concerned with Triebel–Lizorkin–Morrey spaces \({\mathcal {E}}^{s}_{u,p,q}(\Omega )\) of positive smoothness s defined on (special or bounded) Lipschitz domains \(\Omega \subset {{\mathbb {R}}}^{d}\) 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 \({\mathcal {E}}^{s}_{u,p,q}(\Omega )\) using differences of higher order. Special cases include standard Triebel–Lizorkin spaces \(F^s_{p,q} (\Omega )\) and hence classical \(L_p\)-Sobolev spaces \(H^s_p(\Omega )\).