Oscillations and differences in Besov-Morrey and Besov-type spaces

In this paper we investigate Besov-Morrey spaces ${\mathcal{N}}_{u,p,q}^s(\mathrm{\Omega })$ and Besov-type spaces $B_{p,q}^{s,\tau }(\mathrm{\Omega })$ of positive smoothness defined on Lipschitz domains $\mathrm{\Omega }\subset {\mathbb{R}}^d$ as well as on ${\mathbb{R}}^d$. We combine the Hedberg-Netrusov approach to function spaces with distinguished kernel representations due to Triebel, in order to derive novel characterizations of these scales in terms of local oscillations provided that some standard conditions concerning the parameters are fulfilled. In connection with that we also obtain new characterizations of ${\mathcal{N}}_{u,p,q}^s(\mathrm{\Omega })$ and $B_{p,q}^{s,\tau }(\mathrm{\Omega })$ via differences of higher order. By the way we recover and extend corresponding results for the scale of classical Besov spaces $B_{p,q}^s(\mathrm{\Omega })$. Key words: Besov-Morrey space, Besov-type space, Morrey space, Lipschitz domain, oscillations, higher order differences