Research Focus
Harmonic analysis and partial differential equations
I work in harmonic analysis, the field which grew out of the study of Fourier series. What I find particularly fascinating, is the process of discovery and analysis of new mathematical objects, which participate, in one way or another, in the understanding of the physical world. One such specific example are the "iterated Fourier series". Very basic questions such as whether these series converge almost everywhere, turned out to be deeply related to natural phenomena of physical reality. There are also remarkable connections between these objects and other parts of mathematics, such as the theory of multiple zeta functions and its relation to problems from number theory and from physics.
Publications
- Calderón commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions, Rev. Mat. Iberoamericana 30 (2014), 1413–1437.
- Calderón commutators and the Cauchy integral on Lipschitz curves revisited II. The Cauchy integral and its generalizations, Rev. Mat. Iberoamericana 30 (2014), 1089–1122.
- Calderón commutators and the Cauchy integral on Lipschitz curves revisited I. First commutator and generalizations, Rev. Mat. Iberoamericana 30 (2014), 727–750.
- Multi-linear multipliers associated to simplexes of arbitrary length (with T. Tao and C. Thiele); in Advances in Analysis: The Legacy of Elias M. Stein, Princeton University Press, (2014) 346-402.
- Classical and Multilinear Harmonic Analysis, Vol. II (with W. Schlag), Cambridge Studies in Advanced Mathematics 138, Cambridge University Press, xvi + 324 pp., 2013.
- Classical and Multilinear Harmonic Analysis, Vol. I (with W. Schlag), Cambridge Studies in Advanced Mathematics 137, Cambridge University Press, xviii + 370 pp., 2013.
- Paraproducts with flag singularities I. A case study, Rev. Mat. Iberoamericana 23 (2007), 705–742.
- Bi-parameter paraproducts (with J. Pipher, T. Tao, and C. Thiele), Acta Math. 193 (2004), 269–296.
- LpLp estimates for the biest I & II (with T. Tao and C. Thiele), Math. Ann. 329 (2004), 401–426 and 427–461.
- The bi-Carleson operator (with T. Tao and C. Thiele), GAFA 16 (2006), 230–277.
- Multi-linear operators given by singular multipliers (with T. Tao and C. Thiele), J. Amer. Math. Soc. 15 (2002), 469–496.