Svitlana Mayboroda,
Maria Carmen Reguera,
and Alexander Volberg
Original Announcement
This workshop 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 L^{2}(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.