Pointwise Behavior of Sobolev Functions

Abstract

In this chapter the pointwise behavior of Sobolev functions is investigated. Since the definition of a function u ∈ W^k,p(Ω) requires that the k^th-order distributional derivatives of u belong to L^p(Ω), it is therefore natural to inquire whether the function u possesses some type of regularity (smoothness) in the classical sense. The main purpose of this chapter is to show that this question can be answered in the affirmative if interpreted appropriately. Although it is evident that Sobolev functions do not possess smoothness properties in the usual classical sense, it will be shown that if u ∈ W^k,p (Rⁿ when u has derivatives of order k computed in the metric induced by the L^p-norm. That is, it will be shown for all points x in the complement of some exceptional set, there is a polynomial P _x of degree k such that the L^p-norm of the integral average of the remainder |u - p _x | over a ball B(x,r) is o(r^k). Of course, if u were of class C^k then the L^p-norm could be replaced by the sup norm.