Weighted singular integral operators and non-homogenous harmonic analysis

October 10 to October 14, 2011

at the

American Institute of Mathematics, Palo Alto, California

organized by

Svitlana Mayboroda, Maria Carmen Reguera, and Alexander Volberg

This workshop, sponsored by AIM and the NSF, will focus on recent developments on weighted inequalities for singular integral operators and its connection with questions in geometric measure theory and PDE. There are two central problems:

The harmonic analytic question of establishing a characterization of the two-weight inequality for singular integral operators. This characterization would be a profound extension of the famous T1 Theorem of David and Journé. There exist partial solutions to the conjecture assuming extra side conditions on the weights. This partial information has been the source of a quickening series of results.
The geometric measure theoretic question of characterizing rectifiable sets E in terms of boundedness of the Riesz transforms in L²(E). This question is related to many long-standing open problems in the field, including a (higher-dimensional) Painlevé problem regarding the removable singularities of Lipschitz harmonic functions.

In this workshop we aim to overview the competing methods that have been used to study the main problems: the martingale method adapted from the one-weight case, the maximum principle technique, and the Bellman function technique. As well as identify interesting directions for further work in areas that include operator theory, orthogonal polynomials, elliptic pdes, and weighted inequalities of all types in novel settings. The workshop will serve as a time and place to enhance the collaboration of several different groups working at this array of questions.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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