EXTENSION-THEOREMS ON WEIGHTED SOBOLEV SPACES

Let w(i) is-an-element-of A(p)i, 1 less-than-or-equal-to p(i) < infinity for i = 1, 2,..., N. For any unbounded (epsilon, infinity) domain D, by modifying a technique of P. Jones (cf [11]), we show that there exists an extension operator LAMBDA on D such that parallel-to del(k)i LAMBDA f parallel-to L(w)i(p)i(R(n)) less-than-or-equal-to C(i) parallel-to del(k)i f parallel-to L(w)i(p)i (D) for all i where C(i) depends only on epsilon, w(i), k(i), n and max(i)k(i). Moreover, when D is a bounded (epsilon, infinity) domain, a similar but weaker result holds. We also extend P. Jones' result on (epsilon, delta) domains to A(p)-weighted Sobolev spaces. Finally, many applications such as Sobolev interpolation inequalities and Nirenberg-type inequalities on (epsilon, infinity) domains are given.