Michael Cwikel 
Interpolation of Weighted Banach Lattices/A Characterization of Relatively Decomposable Banach Lattices [PDF ebook] 

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 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.