Research

My researches are problems in harmonic analysis that are related to differential geometry, several complex variables and function spaces. My current focus is mostly related to the low regularity problems.

Papers and Preprints (reverse chronological by arXiv submission)

A Linear Operator Bounded in All Besov but not in Triebel-Lizorkin Spaces; (2024), 11 pages, arXiv:2404.05813.
(with Haowen Lu) On Seeley-type Universal Extension Operators for the Upper Half Space; (2022), 19 pages, arXiv:2211.15567.

- - Math. Nachr. 297, 2024, no.3, 811–832, 22 pp.
  - A short slides.

Sobolev and Hölder Estimates for Homotopy Operators of the dbar-Equation on Convex Domains of Finite Multitype; (2022), 34 pages, arXiv:2210.15830.

- - J. Math. Anal. Appl. 538, 2024, no. 2, 128238, 41 pp.
  - Slides.

On Rough Frobenius-type Theorems and Their Hölder Estimates; (2022), 180 pages, PhD Thesis, arXiv:2210.09143.

- - ISBN: 979-8841-74236-4, available at ProQuest LLC.
  - This is a combination of arXiv:2002.07973, 2004.07288, 2105.10120 and 2202.07729 below, along with some new unsubmitted results.
  - Slides for my thesis defense.

Sharp Hölder Regularity for Nirenberg's Complex Frobenius Theorem; (2022), 70 pages. arXiv:2202.07729.

- - To appear in J. Anal. Math., 130 pp.
  - Poster and Slides.

Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains; (2021), 13 pages, arXiv:2112.03996.

- - J. Fourier Anal. Appl. 29, 2023, no. 2, Paper No. 24, 21 pp.

(with Ziming Shi) A Solution Operator for the dbar-Equation in Sobolev Spaces of Negative Index; (2021), 25 pages. arXiv:2111.09245.

- - Trans. Amer. Math. Soc. 377, 2024, no.2, 1111–1139, 39 pp.

(with Ziming Shi) New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications; (2021), 33 pages, arXiv:2110.14477.

- - Math. Nachr. 297, 2024, no. 4, 1407–1443, 37 pp.
  - Slides (short version).

(with Ziming Shi) Sobolev 1/2-Estimate the dbar-Equation on Strictly Pseudoconvex Domains with C^2 boundary; (2021), 32 pages, arXiv:2107.08913.

- - To appear in Amer. J. Math..

(with Brian Street) Improving Regularity of Rough Vector Fields; (2021), 53 pages, arXiv:2105.10120.

- - J. Funct. Anal., 2022, 283, no. 5, Paper No. 109537, 75 pp.

The Frobenius Theorem for Log-Lipschitz Subbundles; (2020), 33 pages, arXiv:2004.07288.

- - J. Geom. Anal. 33, 2023, no. 7, Paper No. 198, 49 pp.
  - Slides.
  - A notes on the sketch and open problem. Used in private communication.

A Counterexample to C^k-regularity for the Newlander-Nirenberg Theorem; (2020), 5 pages, arXiv:2002.07973.

- - Proc. Amer. Math. Soc., 2021, 149(3): 1111--1115.

Miscellaneous

1. Notes on Convexity Estimates on dbar Solution Operators. Used in private communication. (To appeared)
2. A summary note of "Quantitative Bounds in the Nonlinaer Roth Theorem'' (Peluse and Prendiville); joint with Alex McDonald, MLHA 2024 Spring school.
3. The slides (short version) for my talk The Cauchy-Riemann Problems via Extension Operators.
4. Notes of an Introduction to the dbar-Equations, Graduate analysis seminar ---- "topic in dbar-equations and its applications", 2022 Spring.
5. Notes of Wedge Products of Plurisubharmonic Functions. Graduate geometric analysis seminar, 2020 Spring.
6. Notes of "Mass Concentration Phenomena for the $L^2$-Critical Nonlinear Schr\"odinger Equation" (Bégout & Vargas); Graduate analysis seminar, 2019 Fall.
7. Notes for part of "Polynomial Methods in Combinatorics"; Graduate analysis seminar, 2019 Spring.
8. Notes of Newlander-Nirenberg Theorem; Specialty exam, 2018 Fall.
9. Undergraduate Thesis, Summary on Besicovitch sets and Kakeya conjecture, 2018 Spring.
  - There are a lot of typos and grammer errors for the thesis. But I prefer to let it unedited.