Monotone substochastic operators and a new Calderón couple

Karol Leśnik

Studia Mathematica, Tome 231 (2015), p. 21-39 / Harvested from The Polish Digital Mathematics Library

Résumé

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 the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that $(\widetilde{L^{p}},L^{∞})$ is a Calderón couple for 1 ≤ p < ∞, where $\widetilde{L^{p}}$ is the Köthe dual of the Cesàro space $Ces_{p^{\prime }}$ (or equivalently the down space $L^{p^{\prime }}_{↓}$). In particular, $(\widetilde{L¹},L^{∞})$ 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 $(L^{∞}_{↓},L¹)$ is a Calderón couple.

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Publié le : 2015-01-01

Zbl 06446125

EUDML-ID : urn:eudml:doc:285740