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.

Readership

Graduate students and research mathematicians interested in
functional analysis.