Sobolev Embedding Theorem for the Sobolev-Morrey spaces

Abstract

In this paper we prove a Sobolev Embedding Theorem for Sobolev-Morrey spaces. The proof is based on the Sobolev Integral Representation Theorem and on a recent results on Riesz potentials in generalized Morrey spaces of Burenkov, Gogatishvili, Guliyev, Mustafaev and on estimates on the Riesz potentials. We mention that a Sobolev Embedding Theorem for Sobolev morrey spaces had been proved by Campanato, for a subspace of our Sobolev-Morrey space which corresponds to the closure of the set of smooth functions in our Sobolev-Morrey space. The methods of the present paper are considerably different from those of Campanato.