The Concept of a Trace and the Boundedness of the Trace Operator in Banach-Sobolev Function Spaces

Abstract

In this paper a Banach function space (b.f.s. in short) X(Omega) on a n-dimensional bounded domain Omega with Lebesgue measure is considered. The Banach function space X(S) on the (n-1)-dimensional surface S subset of Omega over bar generated by the norm of the space X(Omega) is defined. The Sobolev function space WXm(Omega) is defined using the norm of the space X(Omega), as well as the concept of the trace of an arbitrary function from this space on the surface S. Based on this concept, the space of traces W-X(m)(S) is defined and a characterization of this space is given. It was proved that it is boundedly embedded in the space X(S). Particular cases of a functional space X(Omega) can be Lebesgue spaces, Lebesgue spaces with variable summability exponents, Morrey space, Orlicz space, grand Lebesgue spaces and etc. The obtained results allow us to consider boundary value problems for differential equations in b.f.s. W-X(m)(Omega).