Disjointification inequalities in symmetric quasi-Banach spaces and their applications

Sergey Astashkin, Fedor A. Sukochev and Dmitriy Zanin

Vol. 270 (2014), No. 2, 257–285

DOI: 10.2140/pjm.2014.270.257

Abstract

We demonstrate the relevance of the Prokhorov inequality to the study of Khintchine-type inequalities in symmetric function spaces. Our main result shows that the latter inequalities hold for a pair of quasi-Banach symmetric function spaces $X$ and $Y$ if and only if the Kruglov operator $K$ acts from $X$ to $Y$. We also obtain an extension of von Bahr–Esseen and Esseen–Janson $L_p$-estimates for sums of independent mean zero random variables to the class of quasi-Banach symmetric spaces. In particular, in contrast to the well-known Esseen–Janson theorem, we do not assume that the summands are equidistributed.

Keywords

Kruglov operator, Prokhorov inequality, quasi-Banach spaces

Mathematical Subject Classification 2010

Primary: 46E30

Secondary: 60G50, 46B09

Milestones

Received: 9 December 2012

Accepted: 23 April 2014

Published: 22 August 2014