We prove L p estimates for a two-dimensional bilinear operator of paraproduct type. This result answers a question posed by Demeter and Thiele in [3].

PACS numbers: 71.15.Qe, 71.45.Gm Appendix A: Integral representation of cumulant seriesThe purpose of this appendix is to provide a rigorous mathematical foundation for the transformation (37) and its inverse (39) of the main text. We omit subscripts and superscripts and write simply ρ and C. Our goal is to define the mapping ρ(ν) → C(t), specify its domain and range, prove that it is one-to-one and onto, and describe its inverse map C(t) → ρ(ν). Moreover, at the end of the section we restrict our attention to a rather generic particular case that is sufficient for the application explained in the main part of this paper.Let M + (R) and M(R) respectively denote the space of finite positive measures and the space of finite signed measures on the real line (cf. Sections 1.2 and 4.1 in [1]). Indeed, an element of M(R) is simply a difference of two elements from M + (R). This space contains all absolutely integrable real-valued functions, but it also contains the "true" measures, such as absolutely summable linear combinations of Dirac delta-functions at certain points of R. For any finite signed measure ρ ∈ M(R) we interpret (37)of the main text asThis transformation is well-defined since the function (e −iνt − 1 + iνt)/ν 2 can be extended at ν = 0 in a way that it becomes a bounded continuous function on the whole real line and thus it can be integrated against the measure ρ. The basics of integration theory with respect to abstract measures can be found in the classical textbooks on measure theory, such as [1] and [2]. Let us remark that integration with respect to a general measure ρ is usually written with ρ(ν)dν replaced by dρ(ν) or ρ(dν) in mathematical literature in order to emphasize that ρ need not have a density (i.e. it could be a true measure). We will stick to the former notation, since it is also common to understand signed (or even complex) measures as particular cases of distributions (cf. Section 6.11 in [3]). * Corresponding author. Email: branko@ifs.hr Differentiation of (A1) is justified using the dominated convergence theorem (cf. Theorem 2.27(b) from [2]) and it yieldsĊ In the literature on probability theory (such as Chapter 3 of [7]), ρ is usually called the characteristic function of ρ, even though the normalization is slightly different there. Substituting t = 0 into (A1) and (A2) givesFrom (A3)-(A6) we see that an equivalent way of expressing C in terms of ρ iswith the usual convention thatIt is easy to see that C(t) is always two times continuously differentiable, but it is not true that every such function can be obtained from some ρ ∈ M(R), as we shall soon see.Now we claim that the transform defined by (A1) is a one-to-one map from M(R) to some collection of twice continuously differentiable functions, i.e. that different measures ρ map to different functions C. In order to verify that we assume ρ 1 → C and ρ 2 → C. From (A3) we get2 so the well-known fact that the Fourier-Stieltjes transform ρ → ρ is one-to-one (see Subsection VI. for any choice of points t 1 , . . . , t n ∈ R an...

Abstract. We prove L p estimates for a class of two-dimensional multilinear forms that naturally generalize (dyadic variants of) both classical paraproducts and the twisted paraproduct introduced in [5] and studied in [1] and [6]. The method we use builds on the approach from [6] and we present it as a rather general technique for proving estimates on dyadic multilinear operators. In the particular application to "generalized paraproducts" this method is combined with combinatorics of integer partitions.

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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.

We prove L p bounds for the truncated simplex Hilbert transform which grow with a power less than one of the truncation range in the logarithmic scale.

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