Singular Brascamp–Lieb inequalities with cubical structure

Abstract

We prove a singular Brascamp-Lieb inequality, stated in Theorem 1, with a large group of involutive symmetries.

Boxes, extended boxes and sets of positive upper density in the Euclidean space

“…We finalise the paper with the proof of the remaining analytical result. As we will soon see, the case when all dimensions d i are equal will be an easy consequence of the main result from the paper by Thiele and one of the present authors [7]. We will spend just a slight additional effort to reduce the case of possibly different dimensions d i to the very same result.…”

“…This method reduces Theorems 1-3 to boundedness of certain multilinear singular integral operators. In order to obtain bounds for these operators we invoke the main result from the recent paper by Thiele and one of the present authors [7], which in turn uses techniques gradually developed in a series of papers including [3][4][5][6]11].…”

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

“…First, quite often dyadic models help in developing the techniques that are used later to approach the original, continuous-type problems. The reader can compare the present paper with the work of Durcik and Thiele [11], which is the current state-of-the-art on the continuous singular entangled forms. Second, in some applications it is possible to transfer an estimate easily from dyadic to continuous setting; see [13] and [10].…”

“…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].…”

Averages of simplex Hilbert transforms

“…We currently do not have the estimate (1.10) for any n ≥ 4 (for any tuple of exponents (p j ) j ). This problem is closely related to the question of extending the range of exponents for the twisted paraproduct (1.6), as well as proving L p estimates for the integrals studied in [DT18], in the case when the index set is an arbitrary subset of the cube in R n . However, in §2 we show that (1.2) holds provided that (1.10) holds.…”