Monotone substochastic operators and a new Calderon couple

Abstract

An important result on submajorization, which goes back to Hardy, Littlewood and P\'olya, states that $b\preceq a$ 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 Calder\'on couple for $1\leq p<\infty $, where $\widetilde{L^{p}}$ is the K\"othe dual of the Ces\`aro space $Ces_{p'}$ (or equivalently the down space $L^{p'}_{\downarrow}$). In particular, $(\widetilde{L^1},L^{\infty})$ is a Calder\'on couple and this complements the result of [MS06] where it was shown that $(L^{\infty}_{\downarrow},L^{1})$ is a Calder\'on couple.