Monotone substochastic operators and a new Calderon couple

TL;DR

This paper proves that under monotonicity assumptions, submajorization relations can be realized with monotone doubly stochastic matrices, and applies this to establish new Calderón couple results involving Cesàro and down spaces.

Contribution

It introduces monotone substochastic operators for submajorization and demonstrates that certain pairs of function spaces form Calderón couples, extending previous results.

Findings

Monotone matrices can realize submajorization under monotonicity assumptions.

The pair (L^p, L^\u221e) forms a Calderón couple for 1  p <  .

(L^1, L^) is a Calderón couple, complementing prior work.

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.