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

  • Some key words of my research: Fourier restriction estimates, polynomial methods, Falconer distance set problem, Kakeya conjecture, unit distance problem, decoupling, projection theory, fractal measures, finite point configurations, sparse domination, weighted norm inequalities, multilinear spherical maximal functions, nonlinear Schrödinger equation, random data theory, Strichartz estimates, rough singular integrals, Bochner-Riesz multipliers, Radon transforms, multiparameter 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 CAREER grant DMS #2142221 and standard grant DMS #2055008.

Publications and Preprints

  1. An $L^{3/2}$ $SL_2$ Kakeya maximal inequality, with John Green and Terence L. J. Harris, preprint (2023). HAL
  2. Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets, with Tainara Borges and Benjamin Foster, preprint (2023). arXiv
  3. Weighted refined decoupling estimates and application to Falconer distance set problem, with Xiumin Du, Kevin Ren, and Ruixiang Zhang, preprint (2023). arXiv
  4. New improvement to Falconer distance set problem in higher dimensions, with Xiumin Du, Kevin Ren, and Ruixiang Zhang, preprint (2023). arXiv
  5. A singular variant of the Falconer distance problem, with Tainara Borges and Alex Iosevich, preprint (2023). arXiv
  6. On a free Schrödinger solution studied by Barceló-Bennett-Carbery-Ruiz-Vilela, with Xiumin Du, Hong Wang, and Ruixiang Zhang, preprint (2023). arXiv
  7. Sparse bounds for the bilinear spherical maximal function, with Tainara Borges, Benjamin Foster, Jill Pipher, and Zirui Zhou, J. London Math. Soc. 107 (2023), 1409-1449. DOI: 10.1112/jlms.12715. arXiv
  8. On the multiparameter Falconer distance problem, with Xiumin Du and Ruixiang Zhang, Trans. Amer. Math. Soc. 375 (2022), 4979-5010. DOI: 10.1090/tran/8667. arXiv
  9. An improved result for Falconer’s distance set problem in even dimensions, with Xiumin Du, Alex Iosevich, Hong Wang, and Ruixiang Zhang, Math. Ann. 380 (2021), 1215-1231. DOI: 10.1007/s00208-021-02170-1. arXiv
  10. Finite point configurations and the regular value theorem in a fractal setting, with Krystal Taylor, Indiana Univ. Math. J. 71 (2022), no. 4, 1707-1761. DOI: 10.1512/iumj.2022.71.9054. arXiv
  11. 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
  12. Weighted estimates of singular integrals and commutators in the Zygmund dilation setting, with Xuan Thinh Duong, Ji Li, Jill Pipher, and Brett D. Wick, preprint (2019). arXiv
  13. 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
  14. 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
  15. 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
  16. 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. 143 (2021), no. 1, 175-211. DOI: 10.1353/aim.2021.0005. arXiv
  17. 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
  18. Two-weight inequalities for multilinear commutators, with Ishwari Kunwar, New York J. Math. 24 (2018), 980-1003. arXiv
  19. 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
  20. 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
  21. 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
  22. A cone restriction estimate using polynomial partitioning, with Hong Wang, J. Eur. Math. Soc. 24 (2022), no. 10, 3557-3595. DOI:10.4171/JEMS/1168. arXiv
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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
  31. Multi-parameter singular integral operators and representation theorem, Rev. Mat. Iberoam. 33 (2017), no. 1, 325-350. DOI: 10.4171/RMI/939. arXiv
  32. Upper bound for multi-parameter iterated commutators, with Laurent Dalenc, Publ. Mat. 60 (2016), 191-220. DOI: 10.5565/PUBLMAT_60116_07. arXiv
  33. 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), Tainara Borges (Brown), Wei Chen (YZU), Laura Cladek (ANU), José M. Conde-Alonso (UAM), Amalia Culiuc (Amherst), Laurent Dalenc, Francesco Di Plinio (Università di Napoli Federico II), Xiumin Du (Northwestern), Xuan Thinh Duong (Macquarie), Chenjie Fan (CAS), Benjamin Foster (Stanford), John Green (UPenn), Larry Guth (MIT), Terence Harris (UW-Madison), Alex Iosevich (Rochester), Ishwari Kunwar (Fort Valley State), Michael Lacey (Georgia Tech), Ji Li (Macquarie), Kangwei Li (Tianjin U), Henri Martikainen (WUSTL), Stefanie Petermichl (JMU Würzburg), Jill Pipher (Brown), Kevin Ren (Princeton), Guillermo Rey, Gigliola Staffilani (MIT), Elizabeth Strouse (U Bordeaux), Krystal Taylor (Ohio State U), Emil Vuorinen (Lund U), Hong Wang (NYU), Brett Wick (WUSTL), Bobby Wilson (U Washington), Dongyong Yang (Xiamen U), Ruixiang Zhang (UC Berkeley), Zirui Zhou (UC Berkeley)

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