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, distance set problems, singular integral theory, and sparse domination estimates. I'm also interested in applications of harmonic analysis in nonlinear dispersive PDE, elliptic PDE, combinatorics, and geometric measure theory.

• Some key words of my research: Fourier restriction estimates, polynomial methods, Falconer's distance set problem, unit distance problem, decoupling, nonlinear Schrödinger equation, random data theory, Strichartz estimates, sparse domination, weighted norm inequalities, rough singular integrals, Bochner-Riesz multipliers, Radon transforms, multi-parameter singular integral operators, T1/Tb theorems, Zygmund dilations, 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.
• My research is partially sponsored by the National Science Foundation DMS #1854148.

Preprints

1. An improved result for Falconer's distance set problem in even dimensions, with Xiumin Du, Alex Iosevich, Hong Wang, and Ruixiang Zhang, submitted (2020). arXiv
2. Finite point configurations and the regular value theorem in a fractal setting, with Krystal Taylor, submitted (2020). arXiv
3. Weighted estimates of singular integrals and commutators in the Zygmund dilation setting, with Xuan Thinh Duong, Ji Li, Jill Pipher, and Brett D. Wick, submitted (2019). arXiv
4. A cone restriction estimate using polynomial partitioning, with Hong Wang, submitted (2018). pdf

Publications

1. 2D-defocusing nonlinear Schrödinger equation with random data on irrational tori, with Chenjie Fan, Gigliola Staffilani, and Hong Wang, Stoch PDE: Anal Comp (2020). DOI: 10.1007/s40072-020-00174-7. arXiv
2. Sparse domination and the strong maximal function, with Alex Barron, José M. Conde-Alonso, and Guillermo Rey, Adv. Math. 345 (2019), 1-26. DOI: 10.1016/j.aim.2019.01.007. arXiv
3. On Falconer's distance set problem in the plane, with Larry Guth, Alex Iosevich, and Hong Wang, Invent. Math. 219 (2020), 779-830. DOI: 10.1007/s00222-019-00917-x. arXiv
4. Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type, with Wei Chen, Amalia Culiuc, Francesco Di Plinio, and Michael Lacey, Rev. Mat. Iberoam., to appear (2018). arXiv
5. Weighted restriction estimates and application to Falconer distance set problem, with Xiumin Du, Larry Guth, Hong Wang, Bobby Wilson, and Ruixiang Zhang, Amer. J. Math., to appear (2018). arXiv
6. Commutators of multi-parameter flag singular integrals and applications, with Xuan Thinh Duong, Ji Li, Jill Pipher, and Brett D. Wick, Anal. PDE, 12 (2019), no. 5, 1325-1355. DOI: 10.2140/apde.2019.12.1325. arXiv
7. Two-weight inequalities for multilinear commutators, with Ishwari Kunwar, New York J. Math., to appear (2017). arXiv
8. 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
9. Bilinear representation theorem, with Kangwei Li, Henri Martikainen and Emil Vuorinen, Trans. Amer. Math. Soc. 371 (2019), no. 6, 4193-4214. DOI: 10.1090/tran/7505. arXiv
10. Sparse domination of Hilbert transforms along curves, with Laura Cladek, Math. Res. Lett. 25 (2018), no. 2, 415-436. DOI: 10.4310/MRL.2018.v25.n2.a4. arXiv
11. 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
12. 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
13. 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. 28 (2018), no. 3, 2558-2601. DOI: 10.1007/s12220-017-9920-2. arXiv
14. Domination of multilinear singular integrals by positive sparse forms, with Amalia Culiuc and Francesco Di Plinio, J. London Math. Soc. 98 (2018), no. 2, 369-392. DOI: 10.1112/jlms.12139. arXiv
15. 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
16. A modulation invariant Carleson embedding theorem outside local L^2, with Francesco Di Plinio, J. Anal. Math. 135(2) (2018), 675-711. DOI: 10.1007/s11854-018-0049-4. arXiv
17. Banach-valued multilinear singular integrals, with Francesco Di Plinio, Indiana Univ. Math. J. 67 (2018), no. 5, 1711-1763. DOI: 10.1512/iumj.2018.67.7466. arXiv
18. 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
19. Multi-parameter singular integral operators and representation theorem, Rev. Mat. Iberoam. 33 (2017), no. 1, 325-350. DOI: 10.4171/RMI/939. arXiv
20. Upper bound for multi-parameter iterated commutators, with Laurent Dalenc, Publ. Mat. 60 (2016), 191-220. DOI: 10.5565/PUBLMAT_60116_07. arXiv
21. 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

Collaborators

Alex Barron (UIUC), Wei Chen (YZU), Laura Cladek (UCLA), José M. Conde-Alonso (UAM), Amalia Culiuc (Amherst), Laurent Dalenc, Francesco Di Plinio (WUSTL), Xiumin Du (U Maryland), Xuan Thinh Duong (Macquarie), Chenjie Fan (U Chicago), Larry Guth (MIT), Alex Iosevich (Rochester), Ishwari Kunwar (Fort Valley State), Michael Lacey (Georgia Tech), Ji Li (Macquarie), Kangwei Li (BCAM), Henri Martikainen (U Helsinki), Stefanie Petermichl (JMU Würzburg), Jill Pipher (Brown), Guillermo Rey, Gigliola Staffilani (MIT), Elizabeth Strouse (U Bordeaux), Krystal Taylor (Ohio State U), Emil Vuorinen (Lund U), Hong Wang (IAS), Brett Wick (WUSTL), Bobby Wilson (U Washington), Dongyong Yang (Xiamen U), Ruixiang Zhang (U Wisconsin-Madison)

Ph.D. dissertation

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

Slides

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