Norm-variation of ergodic averages with respect to two commuting transformations

Polona Durcik, Vjekoslav Kovač, Kristina Ana Škreb, Christoph Thiele

Published 2016-03-02Version 1

In this paper we study double ergodic averages with respect to two general commuting transformations and establish quantitative results on their convergence in the norm. In particular we estimate the number of $\varepsilon$-fluctuations in the $L^2$ norm of the sequence of ergodic averages by $\varepsilon^{-\alpha}$ for $\alpha>8$. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the triangular Hilbert transform. $L^p$ estimates for the triangular Hilbert transform remain an open problem.