Norm variation of ergodic averages with respect to two commuting transformations

Abstract

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.

On Side Lengths of Corners in Positive Density Subsets of the Euclidean Space

“…It remains to consider the form associated with the multiplier symbol m [2] , which does not vanish on ξ + η = 0. This part of the proof can be compared with Section 5 in [5]. In the onedimensional case [5], the multiplier was symmetrized to become constant on the axis ξ +η = 0.…”

“…We begin by stating an "integration by parts" lemma, which will be used several times in the proof of Theorem 3. Its one-dimensional variant can be found in [4] or [5], but we prefer to give a self-contained proof. For real-valued functions ψ, ϕ ∈ S(R d ) and F ∈ S(R 2d ) we define the singular integral form…”

“…This is the easier term, as it vanishes on the plane ξ + η = 0, which brings useful cancellation to our form. The remaining part of the proof related to m [1] can be compared with Sections 3 and 4 in [5].…”

“…Only recently the techniques required for bounding the form in Theorem 3 were developed as byproducts of the papers [5] and [6], both of which are primarily concerned with unrelated problems. Indeed, Theorem 3 can be viewed as a higher-dimensional variant of an auxiliary estimate from [5], which established a norm-variation bound…”

“…In Section 2 we give a detailed self-contained proof of Theorem 3. Unlike in [5], where the proof of a special case was given, we do not need any finer control of the constant C here, and are able to make use of further ideas from [6]. Section 3 contains the predominantly combinatorial part of the proof: we derive Theorem 2 from Theorem 3 by mimicking the steps from [3].…”