Power-type cancellation for the simplex Hilbert transform

Abstract

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.

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

“…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…”

“…Inequality (1.4) in turn proved a quantitative result on the convergence of ergodic averages with respect to two commuting transformations. Moreover, the paper [6] studied multilinear analogs of (1.3), with a more modest goal of proving boundedness with constants growing like (log(R/r)) 1−ǫ as R/r → ∞, where the integration variable u is now restricted to intervals [−R, −r] and [r, R] for 0 < r < R. Interestingly, early instances of the method used for solving these problems were devised for bounding significantly less singular variants of the operator (1.3), such as…”

“…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].…”

“…Even though we closely follow the basic steps from [3], the proof diverges at the part relying on harmonic analysis. We need to apply a higher-dimensional variant of a multilinear estimate from [5], which we establish using the techniques from [5] and [6].…”

T(1) theorem for dyadic singular integral forms associated with hypergraphs

“…Entangled multilinear singular integral forms have been studied by several authors over the last ten years; see the papers by Kovač [12], [13], Kovač and Thiele [16], Durcik [3], [4], and Durcik and Thiele [11]. They recently found applications in ergodic theory [14], [8], in arithmetic combinatorics [6], [7], to stochastic integration [15], and within the harmonic analysis itself [9], [10]. Therefore, it would be useful to have a reasonably general theory establishing (or characterizing) L p bounds for these objects.…”