Sobolev–Morrey spaces via Muramatu’s integral formula

Most theorems in Sobolev spaces of positive integer order can be carried over to Sobolev–Morrey spaces by Muramatu’s integral formula if we adjust for Morrey spaces the basic tools such as Hölder’s inequality, Minkowski’s inequality and Young’s inequality. In some theorems the boundedness of the Hardy–Littlewood maximal operator in Morrey spaces plays a crucial role. We show the results corresponding to the interpolation inequality, the embedding into Hölder–Zygmund spaces, the Sobolev inequality, the refined Sobolev inequality involving the homogeneous Besov norm of negative order, the embedding into BMO spaces for the critical case, Trudinger’s inequality for the critical case, the Gagliardo–Nirenberg inequality and its generalization which replaces the [Formula: see text] norm with the BMO norm or the homogeneous Besov norm of order zero, the Brezis–Gallouet–Wainger inequality, and the Brezis–Wainger inequality.