Monotone substochastic operators and a new Calderón couple

Karol Leśnik¹

¹ Institute of Mathematics Poznań University of Technology Piotrowo 3a 60-965 Poznań, Poland

Studia Mathematica, Tome 227 (2015) no. 1, pp. 21-39

Résumé

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.

Détail

DOI : 10.4064/sm227-1-2

Keywords: important result submajorization which goes back hardy littlewood lya states preceq only there doubly stochastic matrix prove under monotonicity assumptions vectors matrix may chosen monotone result applied widetilde infty calder couple leq infty where widetilde dual ces space mathop ces nolimits equivalently down space downarrow particular widetilde infty calder couple which gives positive answer question sinnamon complements result masty sinnamon infty downarrow calder couple