# Interpolation of Weighted Banach Lattices/A Characterization of Relatively Decomposable Banach Lattices

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*Michael Cwikel; Per G. Nilsson; Gideon Schechtman*

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.

#### Table of Contents

# Table of Contents

## Interpolation of Weighted Banach Lattices/A Characterization of Relatively Decomposable Banach Lattices

- Table of Contents v6 free
- Interpolation of weighted Banach lattices 18 free
- 0. Introduction 411
- 1. Definitions, terminology and preliminary results 1118
- 2. The main results 3542
- 3. A uniqueness theorem 4148
- 4. Two properties of the K-functional for a couple of Banach lattices 4653
- 5. Characterizations of couples which are uniformly Calderon-Mityagin for all weights 5562
- 6. Some uniform boundedness principles for interpolation of Banach lattices 7077
- 7. Appendix: Lozanovskii's formula for general Banach lattices of measurable functions 9198
- References 98105

- A characterization of relatively decomposable Banach lattices 106113