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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.
The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate
bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.
We study double averages along orbits for measure-preserving actions of
, the direct sum of countably many copies of a finite abelian group
. We show an
norm-variation estimate for these averages, which in particular re-proves their convergence in
for any finite
and for any choice of two
functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.
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