Oscillation in ergodic theory: Higher dimensional results
In this paper we continue our investigations of square function inequalities. The results in  are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments. The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed.
Israel Journal of Mathematics
Jones, R., Rosenblatt, J., & Wierdl, M. (2003). Oscillation in ergodic theory: Higher dimensional results. Israel Journal of Mathematics, 135, 1-27. https://doi.org/10.1007/BF02776048