Below, I provide a research summary with link to most preprint versions and some context. Other ways to look at my papers are my arXiv feed, my Mathscinet, my ResearchGate and my pages.

LIST OF COAUTHORS AND RELATED RESEARCHERS: an unusual way to find out if we have common interests or are academically related/close.

  • COAUTHORS: Claude Bardos, Mickael Chekroun, Amalia V. Culiuc, Ciprian Demeter, Yen Q. Do, Gregory Duane, Nathan Glatt-Holtz, Andrei Lerner, Yumeng Ou, Vittorino Pata, Roger Temam, Christoph Thiele, Gennady Uraltsev, Sergey Zelik
  • RELATED RESEARCHERS:  Michael Bateman, Dmitry Bilyk, Antoine Choffrut, Monica Conti, Michele Coti Zelati, Yen Do, Noah Graham, Allan Guth, Ben Krause, Michael Lacey, Camil Muscalu, Diogo Oliveira e Silva, Yannis Parissis, Jill Pipher (my unofficial mentor here at Brown), Madalina Petcu, Maria Carmen Reguera, Keith Rogers, Prabath Silva (my academic brother), Stefan Steinerberger, Alexander Volberg, Djoko Wirosoetisno

My two main areas of interest are harmonic analysis and partial differential equations.

In particular, I work in the branch of harmonic analysis called time-frequency analysis, whose central object of study are linear and multilinear singular integrals exhibiting some sort of modulation invariance. Prime examples are Carleson's maximal Fourier partial sums operator, which controls pointwise a.e. convergence of Fourier series for L^p functions, and the bilinear Hilbert transform, a modulation invariant bilinear singular integral related to Calderon's first commutator.
  In the articles joint with C. Thiele and joint with C. Demeter, and in (10) and (11) by myself, we study the behavior of these modulation invariant singular integrals near or at the boundary of the known range of exponents for L^p bounds. In the joint article with A. Lerner, we obtain sharp A_p weighted bounds for Carleson like operators by relating the weighted behavior to the weak type L^p behavior with p close to 1, by means of Andrei's local mean oscillation decomposition technology.
  Surprisingly, Carleson's operator has a deep connection to the problem of differentiation of L^2 functions along smooth vector fields in the plane. In this joint work with C. Demeter, we obtain sharp or almost sharp bounds for the maximal directional Hilbert transform, which is related to both previously objects, in terms of the number of directions.

I also work in PDE, with particular attention to fluid mechanics and to elliptic theory in nonsmooth domains. The articles joint with C. Bardos and R. Temam, and with R. Temam deal with well-posedness of 2D Euler in domains with corners. Especially in the second paper, new elliptic regularity results near L^infinity are obtained, and A_p weights play a role!
  Most of my earlier research deals with the asymptotic behavior of dissipative dynamical systems arising from PDEs of viscoelasticity ((2) and (3) joint with V. Pata, and (1) joint with V. Pata and S. Zelik) and hereditary heat conduction models (with M. Chekroun, N. Glatt-Holtz and V. Pata). In joint work (5), (6) with G. Duane and R. Temam, we propose a new dynamical framework for equations with explicit, possibly singular time dependent terms acting at a functional level. This is inspired by a model relativistic equation called the oscillon equations. See an oscillon here.