On a biparameter maximal multilinear operator

摘要

It is well-known that estimates for maximal operators and questions of pointwise convergence are strongly connected. In recent years, convergence properties of so-called 'non-conventional ergodic averages' have been studied by a number of authors, including Assani, Austin, Host, Kra, Tao, etc. In particular, much is known regarding convergence in L-2 of these averages, but little is known about pointwise convergence. In this spirit, we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator (over R, thanks to a transference principle). Lacey in [15] and Demeter, Tao, and Thiele in [6] have studied maximal multilinear operators previously; however, the maximal operator we develop has a novel biparameter structure which has not been previously encountered and cannot be estimated using their techniques. We will carve this biparameter maximal multilinear operator using a certain Taylor series and produce non-trivial Holder-type estimates for one of the two "main" terms by treating it as a singular integral, the symbol of which has singular set given by two intersecting planes, similarly to that of the Biest operator, studied by Muscalu, Tao, and Thiele in [24] and [25]. (C) 2014 Elsevier Inc. All rights reserved.