Monotone substochastic operators and a new Calderon couple

Abstract

An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that $b⪯a$ if and only if there is a doubly stochastic matrix $A$ such that $b=Aa$. We prove that under monotonicity assumptions on vectors $a$ and $b$ respective matrix $A$ may be chosen monotone. This result is then applied to show that $(\widetilde{L^p},L^{\infty })$ is a Calderón couple for $1\le p<\infty$, where $\widetilde{L^p}$ is the Köthe dual of the Cesàro space $Ce{s}_{p^{\prime }}$ (or equivalently the down space $L_{\downarrow }^{p^{\prime }}$). In particular, $(\widetilde{L^1},L^{\infty })$ is a Calderón couple and this complements the result of [MS06] where it was shown that $(L_{\downarrow }^{\infty },L^1)$ is a Calderón couple.

14 pages