Interpolation of Weighted Banach Lattices: A Characterization of Relatively Decomposable 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 Calderon-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 Calderon-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 Calderon'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.