Carleson theorems and multilinear operators

May 18 to May 22, 2015

at the

American Institute of Mathematics, San Jose, California

organized by

Philip Gressman, Victor Lie, Lillian Pierce, and Po Lam Yung

Original Announcement

This workshop will be devoted to a selection of questions at the intersection of Carleson operators and multilinear operators, with an aim of bringing together a group of established experts and young researchers in order to facilitate the exchange of ideas and open the perspective of collaborative work on a set of core problems.

A celebrated theorem of Carleson (1966) states that the Fourier series of any $L^2$ function on the unit circle converges almost everywhere. This result, as extended by Hunt (1968) and Sjolin (1971), can also be stated in terms of the $L^p$ boundedness of a certain maximal operator involving both oscillatory and singular behavior, now known as the Carleson operator. The ideas at the heart of Carleson's method were substantially enriched by the different perspective of C. Fefferman (1973). These two combined approaches later proved to be influential in the seminal work of Lacey and Thiele on the boundedness of the Bilinear Hilbert Transform (1997/1999).

Ultimately, this circle of ideas and methods developed into a new field of mathematics now known as time-frequency analysis. By now, time-frequency has established direct connections with a number of different areas of mathematics such as ergodic theory, scattering theory and maximal multilinear operators. More recently, in a less direct way, further connections were established with other areas, among which we mention additive combinatorics, geometric measure theory, geometric incidence, and algebraic topology.

The main topics for the workshop are:

  1. The polynomial Carleson operator, and its variants, with particular attention to the possibility of approaching higher dimensional problems
  2. The Hilbert transform along vector fields
  3. Multilinear singular integrals
  4. Multilinear oscillatory integrals

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

A discrete Carleson theorem along the primes with a restricted supremum
by  Laura Cladek, Kevin Henriot, Ben Krause, Izabella Laba and Malabika Pramanik,  Int. Math. Res. Not. IMRN 2017, no. 10, 3180–3208  MR3658135
A discrete quadratic Carleson theorem on $l^2$ with a restricted supremum
by  Ben Krause and Michael Lacey
On a trilinear singular integral form with determinant kernel
by  Philip Gressman, Danqing He, Vjekoslav Kovac, Brian Street, Christoph Thiele and Po-Lam Yung,  Proc. Amer. Math. Soc. 144 (2016), no. 8, 3465-3477  MR3503714