Sparse domination of singular integral operators

October 9 to October 13, 2017

at the

American Institute of Mathematics, San Jose, California

organized by

Amalia Culiuc, Francesco Di Plinio, and Yumeng Ou

Original Announcement

This workshop will be devoted to sparse domination of singular integral operators and to its applications in harmonic analysis, partial differential equations, ergodic theory, and other related fields.

The concept of dominating singular integral operators, which are a priori non-local and non-positive, by sparse operators, which are positive sums of local averages, originated in the work of Lerner (2013) as an alternative route to Hytonen's A2 theorem (2012). Since then, sparse domination has become a leading technique not only within Calderon-Zygmund theory, but also in contexts extending well beyond, such as the study of semigroups of operators, Bochner-Riesz type multipliers, matrix-kerneled and nonhomogeneous singular integrals, oscillatory and arithmetic singular integrals, and modulation invariant multilinear singular integrals. For all such classes of operators, sparse estimates have given rise to new weighted bounds, as well as a wealth of open questions and further directions.

It is tempting to conjecture that suitable sparse theorems hold for all operators that are quasi-local, in the sense that at points far away from the support of the input function, the operator is well approximated by maximal averages. This workshop is designed with the intent of bringing together experts in sparse domination for singular integrals with leading specialists in areas where this principle and its consequences could be further explored.

The main topics for the workshop are

• Sharp sparse domination of rough singular integrals, oscillatory integrals, Radon transforms, Bochner-Riesz multipliers.
• Sharp sparse domination of singular integrals in the nonhomogeneous setting.
• A sparse domination principle for multiparameter singular integrals.
• Sparse domination of modulation invariant singular integrals.
• Sparse domination of oscillatory integrals and Radon-type transforms over the integers.
• A multilinear weighted theory for positive sparse forms and a suitable related extrapolation theory.

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:

An endpoint sparse bound for the discrete spherical maximal functions
by  Robert Kesler, Michael T. Lacey, Darío Mena
Lacunary discrete spherical maximal functions
by  Robert Kesler, Michael T. Lacey, Dario Mena
$l^p$-improving inequalities for discrete spherical averages
by  Michael T. Lacey and Robert Kesler
On logarithmic bounds of maximal sparse operators
by  Grigori A. Karagulyan and Michael T. Lacey
Dyadic harmonic analysis and weighted inequalities: the sparse revolution
by  Mar&\acute;ia Cristina Pereyra
Sparse bounds for pseudodifferential operators
by  David Beltran and Laura Cladek