Abstract
Our rst aim of this paper is to dene 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 oper- ator inequalities of a-quadratic variation and of a-conditional quadratic variation for vector-valued tree martingales hold provided 2 a < 1 by means of Marcinkiewicz interpolation theorem. Based on a result of refer- ence (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.
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