Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2012; 49(2): 395-404

Printed March 1, 2012

https://doi.org/10.4134/JKMS.2012.49.2.395

Copyright © The Korean Mathematical Society.

Some remarks on vector-valued tree martingales

Tong-jun He

Fuzhou University

Abstract

Our first aim of this paper is to define maximal operators of $a$-quadratic variation and of $a$-conditional quadratic variation for vector-valued tree martingales and to show that these maximal operators and maximal operators of vector-valued tree martingale transforms are all sublinear operators. The second purpose is to prove that maximal operator inequalities of $a$-quadratic variation and of $a$-conditional quadratic variation for vector-valued tree martingales hold provided $2\leq a< \infty$ by means of Marcinkiewicz interpolation theorem. Based on a result of reference [10] and using Marcinkiewicz interpolation theorem, we also propose a simple proof of maximal operator inequalities for vector-valued tree martingale transforms, under which the vector-valued space is a UMD space.

Keywords: tree martingales, sublinear operator, maximal operator, Marcinkiewicz interpolation

MSC numbers: Primary 60G46, 60G50, 42C10, 43A75