Research

My research interests lie at the interface of Analysis, Partial Differential Equations (PDE), and Geometric Measure Theory (GMT).  I am interested in understanding the exact relations between the geometry of sets, the structure of the underlying operators, and the behavior of solutions.

You can find my publications on arxiv.

Publication

•  Boundary behavior of solutions of elliptic operators in divergence form with a BMO anti-symmetric part  (with Jill Pipher).Comm. Partial Differential Equations, 44, no. 2 (2019): 156-204.

• $\ell^2$ decoupling in R^2 for curves with vanishing curvature (with Chandan Biswas, Maxim Gilula, Jeremy Schwend, and Yakun Xi). Proc. Amer. Math. Soc. 148, no. 5 (2020): 1987-1997. arXiv:1812.04760

•  The Dirichlet problem for elliptic operators having a BMO anti-symmetric part, (with S. Hofmann, S. Mayboroda, J. Pipher). Math. Ann. (2021) DOI: 10.1007/s00208-021-02219-1.  arXiv:1908.08587

• $L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part (with S. Hofmann, S. Mayboroda, J. Pipher).    Math. Z. 301, pages 935–976 (2022). DOI:10.1007/s00209-021-02938-w    arXiv:1908.01030

• Carleson measure estimates for the Green function, (with G. David and S. Mayboroda). Arch. Ration. Mech. Anal. 243, pages1525–1563 (2022).  DOI: 10.1007/s00205-021-01746-0. arXiv:2102.09592

•  Carleson estimates for the Green function on domains with lower dimensional boundaries,  (with G. David and S. Mayboroda). J. Funct. Anal. Volume 283, Issue 5 (2022). DOI: 10.1016/j.jfa.2022.109553  arXiv:2107.08101

• Small $A_\infty$ results for weak Dahlberg-Kenig-Pipher operators in sets with uniformly rectifiable boundaries, (with G. David, S. Mayboroda).  Trans. Amer. Math. Soc. (2023) DOI: 10.1090/tran/9004  arxiv:2207.13602

• Green functions and smooth distances, (with J. Feneuil, S. Mayboroda).  Math. Ann. (2023). DOI: 10.1007/s00208-023-02715-6 arXiv:2211.05318 

• A Green function characterization of uniformly rectifiable sets of any codimension, (with J. Feneuil). Adv. Math.  430, (2023) Paper No. 109220, DOI: 10.1016/j.aim.2023.109220 arXiv:2302.14087, slides