Some integral inequalities with applications to the imbedding of Sobolev spaces defined over irregular domains

Abstract

This paper examines the possibility of extending the Sobolev Imbedding Theorem to certain classes of domains which fail to have the “cone property” normally required for that theorem. It is shown that no extension is possible for certain types of domains (e.g. those with exponentially sharp cusps or which are unbounded and have finite volume), while extensions are obtained for other types (domains with less sharp cusps). These results are developed via certain integral inequalities which generalize inequalities due to Hardy and to Sobolev, and are of some interest in their own right. The paper is divided into two parts. Part I establishes the integral inequalities; Part II deals with extensions of the imbedding theorem. Further introductory information may be found in the first section of each part.