New proofs of the trace theorem of Sobolev spaces

Abstract

We present three new proofs of the trace theorem of $L_{p}$ Sobolev spaces, which do not rely on the theory of interpolation spaces. The first method originates in Morrey’s proof for the Sobolev embedding theorem concerning the Hölder-Zygmund space. The second method is based on Muramatu’s integral formula and the third method is based on an integral operator with Gauss kernel. These methods give unified viewpoints for the proofs of the trace theorem and the Sobolev embedding theorem.