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

Abstract

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. 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 so-called triangular Hilbert transform.

arXiv comment28 pages

References in corpus (10)

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Cited by in corpus (14)

• Some Problems in Harmonic Analysis
• On side lengths of corners in positive density subsets of the Euclidean space
• Power-type cancellation for the simplex Hilbert transform
• A variational restriction theorem
• Singular Brascamp-Lieb inequalities with cubical structure
• A Szemerédi-type theorem for subsets of the unit cube
• Boxes, extended boxes, and sets of positive upper density in the Euclidean space
• Density theorems for anisotropic point configurations
• Norm-variation of cubic ergodic averages
• Convergence of ergodic-martingale paraproducts
• T(1) theorem for dyadic singular integral forms associated with hypergraphs
• The non-resonant bilinear Hilbert--Carleson operator
• Large dilates of hypercube graphs in the plane
• Norm-variation of triple ergodic averages for commuting transformations