5,181 research outputs found
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: vertices of a
fixed -dimensional rectangular box, the same vertices extended with
points completing three-term arithmetic progressions, and the same vertices
extended with 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
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