Research

My research interests are in the areas of classical harmonic analysis and its related fields. I've been working with problems related to Fourier restriction estimates, sparse domination of operators of various types, as well as multi-parameter and multilinear singular integral operators with their Banach-valued extensions. I'm also interested in applications of harmonic analysis in (dispersive or elliptic) PDE.

Some key words of my research: sparse domination, weighted norm inequalities, rough singular integrals, Bochner-Riesz multipliers, Radon transforms, Fourier restriction estimates, polynomial methods, Falconer's distance set problem, decoupling, multi-parameter T1/Tb theorems, representation theorems, dyadic shifts, paraproducts, iterated and higher order commutators, multi-parameter BMO spaces, time-frequency analysis, multilinear singular integrals, UMD spaces, bilinear Hilbert transform, L^p theory for outer measures, Carleson embedding theorems, etc. 

Publications:

  1. A sparse estimate for multisublinear forms involving vector-valued maximal functions, with Amalia Culiuc and Francesco Di Plinio, Bruno Pini Math. Analysis Seminar, p. 168-184, May 2018. arXiv
  2. Bilinear representation theorem, with Kangwei Li, Henri Martikainen and Emil Vuorinen, Trans. Amer. Math. Soc., to appear (2017). arXiv
  3. Sparse domination of Hilbert transforms along curves, with Laura Cladek, Math. Res. Lett., to appear (2017). arXiv
  4. A sparse domination principle for rough singular integrals, with José M. Conde-Alonso, Amalia Culiuc, and Francesco Di Plinio, Anal. PDE, 10 (2017), no. 5, 1255-1284. DOI: 10.2140/apde.2017.10.1255. arXiv
  5. Uniform sparse domination of singular integrals via dyadic shifts, with Amalia Culiuc and Francesco Di Plinio, Math. Res. Lett. 25 (2018), no. 1, 21-42. DOI: 10.4310/MRL.2018.v25.n1.a2. arXiv
  6. Product BMO, little BMO and Riesz commutators in the Bessel setting, with Xuan Thinh Duong, Ji Li, Brett D. Wick, and Dongyong Yang, J. Geom. Anal., to appear (2016). arXiv
  7. Domination of multilinear singular integrals by positive sparse forms, with Amalia Culiuc and Francesco Di Plinio, J. London Math. Soc., to appear (2016). arXiv
  8. Little BMO and Journé commutators, with Stefanie Petermichl, Harmonic analysis, function theory, operator theory and applications, 207-219, Theta Ser. Adv. Math., Theta, Bucharest, 2017. pdf
  9. A modulation invariant Carleson embedding theorem outside local L^2, with Francesco Di Plinio, J. Anal. Math., to appear (2015). arXiv
  10. Banach-valued multilinear singular integrals, with Francesco Di Plinio, Indiana Univ. Math. J., to appear (2015). arXiv
  11. Higher order Journé commutators and characterizations of multi-parameter BMO, with Stefanie Petermichl and Elizabeth Strouse, Adv. Math. 291 (2016), 24-58. DOI: 10/1016/j.aim.2015.12.029. arXiv
  12. Multi-parameter singular integral operators and representation theorem, Rev. Mat. Iberoam. 33 (2017), no. 1, 325-350. DOI: 10.4171/RMI/939. arXiv
  13. Upper bound for multi-parameter iterated commutators, with Laurent Dalenc, Publ. Mat. 60 (2016), 191-220. DOI: 10.5565/PUBLMAT_60116_07. arXiv
  14. A T(b) theorem on product spaces, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6159-6197. DOI: 10.1090/S0002-9947-2015-06246-1. arXiv 

Preprints:

  1. Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type, with Amalia Culiuc, Francesco Di Plinio, and Michael Lacey, submitted (2018). arXiv
  2. Weighted restriction estimates and application to Falconer distance set problem, with Xiumin Du, Larry Guth, Hong Wang, Bobby Wilson, and Ruixiang Zhang, submitted (2018). arXiv
  3. Commutators of multi-parameter flag singular integrals and applications, with Xuan Thinh Duong, Ji Li, Jill Pipher, and Brett D. Wick, submitted (2018). arXiv
  4. Two-weight inequalities for multilinear commutators, with Ishwari Kunwar, submitted (2017). arXiv
  5. A cone restriction estimate using polynomial partitioning, with Hong Wang, submitted (2017). arXiv

Ph.D. dissertation: Multi-parameter commutators and new function spaces of bounded mean oscillation, Brown University (2016). pdf

Slides:

  1. A cone restriction estimate using polynomial partitioning. pdf
  2. Sparse domination of singular integral operators. pdf
  3. Domination of multilinear singular integrals by positive sparse forms. pdf
  4. Banach-valued T(1) type theorems. pdf
  5. Commutators of singular integrals. pdf
  6. T(1) and T(b) theorems on product spaces. pdf