Oscillations and differences in Triebel–Lizorkin–Morrey spaces

In this paper we are concerned with Triebel–Lizorkin–Morrey spaces E u , p , q s ( Ω ) of positive smoothness s defined on (special or bounded) Lipschitz domains Ω ⊂ R d as well as on 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 E u , p , q s ( Ω ) using differences of higher order. Special cases include standard Triebel–Lizorkin spaces F p , q s ( Ω ) and hence classical L p -Sobolev spaces H p s ( Ω ) .