Memoirs of the American Mathematical Society 2003; 127 pp; softcover Volume: 165 ISBN10: 0821833820 ISBN13: 9780821833827 List Price: US$59 Individual Members: US$35.40 Institutional Members: US$47.20 Order Code: MEMO/165/787
 Interpolation of Weighted Banach Lattices It is known that for many, but not all, compatible couples of Banach spaces \((A_{0},A_{1})\) it is possible to characterize all interpolation spaces with respect to the couple via a simple monotonicity condition in terms of the Peetre \(K\)functional. Such couples may be termed CalderónMityagin couples. The main results of the present paper provide necessary and sufficient conditions on a couple of Banach lattices of measurable functions \((X_{0},X_{1})\) which ensure that, for all weight functions \(w_{0}\) and \(w_{1}\), the couple of weighted lattices \((X_{0,w_{0}},X_{1,w_{1}})\) is a CalderónMityagin couple. Similarly, necessary and sufficient conditions are given for two couples of Banach lattices \((X_{0},X_{1})\) and \((Y_{0},Y_{1})\) to have the property that, for all choices of weight functions \(w_{0}, w_{1}, v_{0}\) and \(v_{1}\), all relative interpolation spaces with respect to the weighted couples \((X_{0,w_{0}},X_{1,w_{1}})\) and \((Y_{0,v_{0}},Y_{1,v_{1}})\) may be described via an obvious analogue of the abovementioned \(K\)functional monotonicity condition. A number of auxiliary results developed in the course of this work can also be expected to be useful in other contexts. These include a formula for the \(K\)functional for an arbitrary couple of lattices which offers some of the features of Holmstedt's formula for \(K(t,f;L^{p},L^{q})\), and also the following uniqueness theorem for Calderón's spaces \(X^{1\theta }_{0}X^{\theta }_{1}\): Suppose that the lattices \(X_0\), \(X_1\), \(Y_0\) and \(Y_1\) are all saturated and have the Fatou property. If \(X^{1\theta }_{0}X^{\theta }_{1} = Y^{1\theta }_{0}Y^{\theta }_{1}\) for two distinct values of \(\theta\) in \((0,1)\), then \(X_{0} = Y_{0}\) and \(X_{1} = Y_{1}\). Yet another such auxiliary result is a generalized version of Lozanovskii's formula \(\left( X_{0}^{1\theta }X_{1}^{\theta }\right) ^{\prime }=\left (X_{0}^{\prime }\right) ^{1\theta }\left( X_{1}^{\prime }\right) ^{\theta }\) for the associate space of \(X^{1\theta }_{0}X^{\theta }_{1}\). A Characterization of Relatively Decomposable Banach Lattices Two Banach lattices of measurable functions \(X\) and \(Y\) are said to be relatively decomposable if there exists a constant \(D\) such that whenever two functions \(f\) and \(g\) can be expressed as sums of sequences of disjointly supported elements of \(X\) and \(Y\) respectively, \(f = \sum^{\infty }_{n=1} f_{n}\) and \(g = \sum^{\infty }_{n=1} g_{n}\), such that \(\ g_{n}\ _{Y} \le \ f_{n}\ _{X}\) for all \(n = 1, 2, \ldots\), and it is given that \(f \in X\), then it follows that \(g \in Y\) and \(\ g\ _{Y} \le D\ f\ _{X}\). Relatively decomposable lattices appear naturally in the theory of interpolation of weighted Banach lattices. It is shown that \(X\) and \(Y\) are relatively decomposable if and only if, for some \(r \in [1,\infty ]\), \(X\) satisfies a lower \(r\)estimate and \(Y\) satisfies an upper \(r\)estimate. This is also equivalent to the condition that \(X\) and \(\ell ^{r}\) are relatively decomposable and also \(\ell ^{r}\) and \(Y\) are relatively decomposable. Readership Graduate students and research mathematicians interested in functional analysis. Table of Contents Interpolation of weighted Banach lattices, by Michael Cwikel and Per G. Nilsson  Introduction
 Definitions, terminology and preliminary results
 The main results
 A uniqueness theorem
 Two properties of the \(K\)functional for a couple of Banach lattices
 Characterizations of couples which are uniformly CalderónMityagin for all weights
 Some uniform boundedness principles for interpolation of Banach lattices
 Appendix: Lozanovskii's formula for general Banach lattices of measurable functions
 References
A characterization of relatively decomposable Banach lattices, by Michael Cwikel, Per G. Nilsson and Gideon Schechtman  Introduction
 Equal norm upper and lower \(p\)estimates and some other preliminary results
 Completion of the proof of the main theorem
 Application to the problem of characterizing interpolation spaces
 References
