Bellman function technique for multilinear estimates and an application to generalized paraproducts

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.Comment: 28 pages, 7 figures/diagrams/table

Boundedness of the twisted paraproduct

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].Comment: 21 pages, 2 diagrams, 1 pictur

Sobolev norm estimates for a class of bilinear multipliers

We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing L^p estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential operators of order 0.Comment: 11 page

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

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed $n$-dimensional rectangular box, the same vertices extended with $n$ points completing three-term arithmetic progressions, and the same vertices extended with $n$ points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.Comment: 19 pages; v2: minor changes following referee's repor