In this paper we investigate Besov-Morrey spaces $\mathcal{N}^{s}_{u,p,q}(\Omega)$ and Besov-type spaces $B^{s,\tau}_{p,q}(\Omega)$ of positive smoothness defined on Lipschitz domains $\Omega \subset \mathbb{R}^d$ 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 $\mathcal{N}^{s}_{u,p,q}(\Omega)$ and $B^{s,\tau}_{p,q}(\Omega)$ via differences of higher order. By the way we recover and extend corresponding results for the scale of classical Besov spaces $B^{s}_{p,q}(\Omega)$. Key words: Besov-Morrey space, Besov-type space, Morrey space, Lipschitz domain, oscillations, higher order differences

翻译：本文研究了定义于Lipschitz区域$\Omega \subset \mathbb{R}^d$及$\mathbb{R}^d$上具有正光滑性的Besov-Morrey空间$\mathcal{N}^{s}_{u,p,q}(\Omega)$与Besov型空间$B^{s,\tau}_{p,q}(\Omega)$。我们将Hedberg-Netrusov的函数空间研究方法与Triebel提出的具有显著核表示的理论相结合，在参数满足标准条件的前提下，获得了这些空间尺度基于局部振荡的新刻画。与此同时，我们还通过高阶差分得到了$\mathcal{N}^{s}_{u,p,q}(\Omega)$和$B^{s,\tau}_{p,q}(\Omega)$的新表征。在此过程中，我们恢复并推广了经典Besov空间$B^{s}_{p,q}(\Omega)$尺度上的相应结果。关键词：Besov-Morrey空间，Besov型空间，Morrey空间，Lipschitz区域，振荡，高阶差分