On a biparameter maximal multilinear operator

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 L2L2 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 RR, 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 Hölder-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].