Monotone substochastic operators and a new Calderón couple
Volume 227 / 2015
Abstract
An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, 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 the vectors $a$ and $b$ the 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\leq p<\infty $, where $\widetilde{L^{p}}$ is the Köthe dual of the Cesàro space $\mathop{\rm Ces}\nolimits_{p'}$ (or equivalently the down space $L^{p'}_{\downarrow}$). In particular, $(\widetilde{L^1},L^{\infty})$ is a Calderón couple, which gives a positive answer to a question of Sinnamon [Si06] and complements the result of Mastyło and Sinnamon [MS07] that $(L^{\infty}_{\downarrow},L^{1})$ is a Calderón couple.
