By K.M. Rangaswamy, David Arnold

ISBN-10: 0824797892

ISBN-13: 9780824797898

Includes the complaints of a global convention on abelian teams and modules held lately in Colorado Springs. offers the most recent advancements in abelian teams that experience facilitated cross-fertilization of latest recommendations from varied parts corresponding to the illustration thought of posets, version conception, set thought, and module thought.

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Also Theorem 7 below). Let A a n d / ~ be Banach pairs, and let {po}oeo, {ao}oeo be two families of quasi-concave functions. Set, v ( s , t ) = inf0 p0(,) ~0(t)' then the following two conditions are equivalent for an operator T (cf. [57]) 1 T : ~lpo,i;J ~ Bao,oo;K, 0 ~ 0 K(t, Ta;[~) < r(s,t)J(s,a;fi,), s,t > O. We shall now illustrate these results with some i m p o r t a n t examples. 17) Suppose t h a t fi. is a pair of mutually closed spaces. Then, using Theorem 2, we see that for a _ 0, we have, with equivalence of norms, •0<0<1 (0 -a Ao,1;j) = Apa,1;J.

5) holds, t:] Combining the previous theorem with the characterization of the spaces -Ap,1;J as extrapolation spaces gives (cf. Chapter 2, Theorem 2). To is exact of order O. 2. 6) ) }. By Theorem 2, we have, with p(t) = sup0e(o,x){t~ Proof. tllrj Thus, if we let 1 _ i~f{s0~o(O)} then = Ilf XTr IlfllY~ ' II-f~x j as desired. 2 Applications: Sobolev Imbeddings In this section we consider applications to Sobolev imbedding theorems. Let us record first the following general remark E x a m p l e 17 Let fi~ be an ordered mutually closed pair, and let X be a Banach space.

T : 9v0(+i) -+1 ~'0(/~),V0 e O ': T : +i -+/~ 26 CHAPTER 2. BACKGROUND ON EXTRAPOLATION It turns out that complete families can be characterized by their action on one dimensional spaces. This is the content of the next result. T h e o r e m 7 (cf. To. ;} Proof. 35) and refer to [57] for a proof of the necessity of this condition. ol. 3. 36) gives J(t, Ta;B) < cJ(t,a;A). ) is dense in Ao and A1, we obtain T : A ~ / 3 . To exact of exponent O, is complete. Proof. s and apply the previous theorem to conclude.

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