Introduction to L (p) Sobolev Spaces via Muramatu's Integral Formula

摘要

Muramatu's integral formula is a very useful tool for the study of Sobolev spaces, although this does not seem to be widely recognized. Most theorems in Sobolev spaces can be proved by this formula combined with basic inequalities in analysis, and it is possible to directly treat not only the whole space but also a special Lipschitz domain. In this paper, we present an introduction to L (p) -based Sobolev spaces of integer order by making Muramatu's integral formula play a central role, as Cauchy's integral formula does in complex analysis. The topics we take up are approximation by smooth functions, the interpolation inequality, the Sobolev embedding theorems, the trace theorem, construction of an extension operator, complex interpolation of Sobolev spaces and real interpolation of Sobolev spaces.