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.