WALLIN

There are three new main results in this paper: Theorem 1 on the definition of the trace in Section 2, Theorem 2 on the image of the trace operator in Section 2, and Theorem 3 on the kernel of the trace operator in Section 3. For the application to Dirichlet's problem, see [10].

Notation /? is a bounded open set in R** with boimdary du = Г and а(х,Г) is the distance from x 6 R" to Г. B{x,r) is the closed n-dimensional ball with center x and radius r. The c?-dimensional HausdorfF measure is denoted by m^ (0 < 6? < n); suitably normalized m« coincides with n-dimensional Lebesgue measure and is denoted by m; c?-a.e. means m^-a.e. and a.e. means n-a.e. D^ stands for the partial derivative corresponding to a multi-index j = (ji, ...,7n) with length \j\ = il -h... -fin- 'Pk is the set of all polynomials in n real variables of total degree at most k, Wj[{f2) stands for the Sobolev space of LP{Q) functions with

о

distributional derivatives up to order к in L^{Q) with the usual norm. W\{Q) is the closure of C^{Q) in W^(i?). All functions axe assumed to be real-valued.

2 The trace theorem

2 . 1 We shall first introduce the concepts of <i-set, (б, ^)-domain and sets ing Markov's inequality, and state a theorem by P. Jones (Proposition 3).

Let ^ be a Borel set in R" and rrid the cf-dimensional HausdorfF measure. We call E a d-set, 0 < d < n ([6], Ch. II and VIII), if there exist positive constants ci and C2 so that

cir * * < md{E n B{x, r)) < Cir"^, for x e E, 0 < r < 1.

An open connected subset Ü of R" is an (e, 6)~dornain, £ > 0, oo > 6 > 0 [2], if whenever x^y £ Q and \x — y\< 8^ there is a rectifiable arc у С О with length £{j) joining X to у and satisfying i) %) < |a: - y\/e and

ii ) d(z,y)>e\x - z\\y-z\/\x-y\, for z e j. A closed subset F of R** preserves Markov's inequality ([6], Ch. II) if for every positive integer к there exists a constant с = c{F,n^k) so that

maxlVPI < -maxIPI,

for all P € ^ib and all В = Б(ж,г), x e F, г < 1.

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