We recently sat down with Professor of Mathematics David Cruz-Uribe to discuss two articles that were published earlier this year, both of which were part of an extensive effort to solve what is referred to as the A2 problem in harmonic analysis. The first of these articles was co-authored by Cruz-Uribe and longtime collaborators José María Martell from the Instituto de Ciencias Matematicas in Madrid, Spain, and Carlos Peréz from the Universidad de Sevilla in Sevilla, Spain. The second paper was co-authored by Cruz-Uribe and Kabe Moen, assistant professor of mathematics at the University of Alabama. Please see the bottom of this post for full citations.

Faculty Highlights: What is the A2 problem in harmonic analysis and where did it originate?

David Cruz-Uribe: The A2 problem was first posed in the mid-1990s by Professor Robert Fefferman at the University of Chicago. He asked for the sharpest constant in certain inequalities—more precisely, in weighted Lp norm inequalities for singular integrals. The problem soon expanded to include similar questions for the other kinds of operators in harmonic analysis.

Faculty Highlights: How did the two articles you worked on address this problem?

David Cruz-Uribe: The first paper yielded a very elementary proof of the conjectured result for a special family of singular integrals—the Hilbert transform, Riesz transforms, and the Beurling-Ahlfors transform. These results were previously known, but our approach yielded a unified proof that was considerably simpler than all known proofs. The second paper extended these ideas to a family of operators called commutators.

Faculty Highlights: What was the outcome of these papers?  

Cruz-Uribe: In very basic terms, my colleagues and I vastly simplified what had previously required a 45-page proof into a couple of pages. Though we were not able to refine our argument to get the complete solution, the two papers still marked an advance over what had been done previously.

Faculty Highlights: Has any work been done on this topic since?

David Cruz-Uribe: Yes, shortly after these papers were completed the full A2 conjecture was solved by Tuomas Hytonen of the University of Helsinki using different methods. More recently, however, Andrei Lerner of Bar-Ilan University extended our approach to give a very elegant proof of the conjecture.


David Cruz-Uribe, José María Martell, and Carlos Peréz. “Sharp Weighted Estimates for Classical Operators.” Advances in Mathematics 229, no. 1 (January 15, 2012): 408-441.

David Cruz-Uribe. Kabe Moen. “Sharp Norm Inequalities for Commutators of Classical Operators.” Publicacions Mathematiques 56, no. 1 (2012): 147-190.