My research interests lie in Harmonic Analysis and Partial Differential Equations (PDEs). In my PhD Thesis, together with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi; we studied solvability issues related to Boundary Value Problems (BVPs) for the general second order elliptic operator in the upper half-space in $\mathbb{R}^{n+1}$.
Recently I have also been interested in Geometric Measure Theory and its relationship with PDEs. Together with Simon Bortz, Steve Hofmann, John Hoffman, and Kaj Nyström; we study generalizations of concepts arising in the context of uniform rectifiability to geometric situations different from the Euclidean one studied by Guy David and Stephen Semmes.
Publications:
"Critical Perturbations for Second Order Elliptic Operators. Part 1: Square Function bounds for Layer Potentials" joint with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi. Preprint: https://arxiv.org/abs/2003.02703 Accepted in Analysis and PDE.
"Critical Perturbations for Second Order Elliptic Operators. Part 2: Solvability" joint with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi. Preprint https://arxiv.org/abs/2302.02746 .
"Coronizations and big pieces in metric spaces" joint with Simon Bortz, John Hoffman, Steve Hofmann, and Kaj Nyström. Ann. Inst. Fourier (Grenoble) 72 (2022), no. 5, 2037–2078
"On Big Pieces approximations of parabolic hypersurfaces" joint with Simon Bortz, John Hoffman, Steve Hofmann, and Kaj Nyström. Ann. Fenn. Math. 47 (2022), no. 1, 533–571
"Corona decompositions for parabolic uniformly rectifiable sets" joint with Simon Bortz, John Hoffman, Steve Hofmann, and Kaj Nyström. Submitted. https://arxiv.org/abs/2103.12497
"Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets" joint with Simon Bortz, John Hoffman, Steve Hofmann, and Kaj Nyström. https://arxiv.org/abs/2103.12502
"Parabolic singular integrals with nonhomogeneous kernels" joint with Simon Bortz, John Hoffman, Steve Hofmann, and Kaj Nyström. https://arxiv.org/abs/2103.12830
"Stability of Weighted Norm Inequalities" joint with Michel Alexis, Eric Sawyer, and Ignacio Uriarte-Tuero. https://arxiv.org/abs/2208.08400
"The T1 theorem for the Hilbert transform fails when p is not 2" joint with Michel Alexis, Eric Sawyer, and Ignacio Uriarte-Tuero. Preprint https://arxiv.org/abs/2301.10046
Together with Simon Bortz and Steve Hofmann we also have some lecture notes in IAS/Park City Mathematics series Vol. 27: Harmonic Analysis and Applications, titled "T1 and Tb theorems and applications". You can find the book, with many other interesting lecture notes, here.