SOME REMARKS ON VECTOR-VALUED TREE MARTINGALES

Title & Authors
SOME REMARKS ON VECTOR-VALUED TREE MARTINGALES
He, Tong-Jun;

Abstract
Our first aim of this paper is to define maximal operators a-quadratic variation and of a-conditional quadratic variation for vectorvalued 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 $\small{{\leq}}$ a < $\small{\infty}$ 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;
Language
English
Cited by
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