Quasiconformal mappings and extendability of functions in sobolev spaces

Abstract

Let ~ be an open connected domain in R n, n ~2. If a is a multi-index, a= (ai, ~2 ..... a~)EZ~, the length of a, denoted by [a I, is the integer Xj%~ ~ and D a= (~/~xl) ~" ... (~/~x~)~% A locMly integrable function / on/9 has a weak derivative of order if there is a locally integrable function (denoted by D~ such that fv/( D:cf)dx = (-1) I~j f(D:'/)cfdx for all C ~ functions ~ with compact support in ~. For 1 <p <~ co, k EN, L~(]O) is the Sobolev space of functions having weak derivatives of all orders zr I ~ I ~ k, and satisfying An extension operator on L~(D) is a bounded linear operator A: L~(D) -~ L~(R ~) -= L~ such that A/Iv= / for all/EL~(V). We say that O is an extension domain for Sobolev spaces (E.D.S.) if whenever 1 ~<p~< c~, kEN, there is an extension operator for L~(O).(~)The following theorem is by now well known.(1) ~I.S.F. grant MCS-7905036.(2) We do not require A to be an extension operator also ort L~(~) for m<k. In fact, the one which will be constructed does not have that property.

A Sobolev Extension Domain That is Not Uniform

“…Using Hölder's inequality and a variant of the classical Poincare-Sobolev inequality (see [J,Lemma 2.2] or [HK1,Lemma 4.2]), we deduce that…”

“…But the question of whether the implication QED⇒Loewner can be reversed remains open. However, all other implications in (1.1) can not be reversed in general, as illustrated by numerous examples (see [GM,HK1,HK2,J]). Therefore it is important to seek for conditions under which the implications in (1.1) can be reversed.…”

Besov spaces on domains in ${\bf R}\sp d$

“…We shall establish our results for two important classes of nonsmooth domains: the Lipschitz graph domains, and the (e, S) domains introduced by Jones [6]. We begin in §4 with the case of Lipschitz graph domains since the geometric arguments in this case are the most obvious.…”

“…We shall indicate in this section the adjustments required in §4 to execute the extension theorem for (e, S) domains as introduced by P. Jones [6]. Such domains include as special cases the minimally smooth domains in the sense of Stein [S, p. 189].…”

“…Theorem 6.7. Let Q be a minimally smooth domain in Rd, and 1 < p < oo, 0 < a, then WpaiQ) = Bp(Lp(Q)) and there exist positive constants cx, c2 independent of f so that (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23) dll/II^O) < 11/11^(1,(0)) < c2\\f\\ma).…”