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Interpolation of Weighted Banach Lattices/A Characterization of Relatively Decomposable Banach Lattices
Michael Cwikel, Technion-Israel Institute of Technology, Haifa, Israel, Per G. Nilsson, Lund, Sweden, and Gideon Schechtman, Weizmann Institute of Science, Rehovot, Israel
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Memoirs of the American Mathematical Society
2003; 127 pp; softcover
Volume: 165
ISBN-10: 0-8218-3382-0
ISBN-13: 978-0-8218-3382-7
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 Calderón-Mityagin 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 Calderón-Mityagin 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 above-mentioned $$K$$-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.

Graduate students and research mathematicians interested in functional analysis.

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 Calderón-Mityagin 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