Bi-Parameter Maximal Multilinear Operators

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, and so on. In particular, much is known regarding convergence in L2 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 [17] and Demeter, Tao, and Thiele in [8] have studied maximal multilinear operators previously; however, the maximal operator we develop has a novel bi-parameter structure which has not been previously encountered and cannot be estimated using their techniques. We will carve this bi-parameter maximal multilinear operator using a certain Taylor series and produce non-trivial H¨lder-type estimates for the "main" terms by o treating them as singular integrals whose symbol's singular set is similar to that of the Biest operator, studied by Muscalu, Tao, and Thiele in [26] and [27]. Modulo further work to estimate certain error terms coming from the Taylor series which a priori seem to be well-behaved, this will allow us to estimate the full bi-parameter maximal multilinear operator