Research

Books

  "Harmonic measure from a geometric and an analytic point of view", with C. Kenig and L. Lanzani, University Lectures Series 35, 2005, ISBN 0-8218-2728-6.

  "An introduction to the Heisenberg Group and to the sub-Riemannian Isoperimetric problem", with D. Danielli, S. Pauls and J. Tyson. Birkhäuser , "Progress in Mathematics". 2007, Approx. 230 p., Hardcover, ISBN-10: 3-7643-8132-9 ISBN-13: 978-3-7643-8132-5

Preprints and Publications

•  Sharp Hölder regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces (joint with Ryan Gibara, Riikka Korte and Nageswari Shanmugalingam). Under review. arXiv:2505.14950 

• Fractional p-Laplacians via Neumann problems in unbounded metric measure spaces (joint with Ryan Gibara, Riikka Korte and Nageswari Shanmugalingam). Under review. Preprint ArXiv:2410.18883v1

• Green function in metric measure spaces (joint with Mario Bonk and Xiaodan Zhou).  Under review. Preprint ArXiv:2211.11974

• Neural networks for threshold dynamics reconstruction (joint with Elisa Negrini, Almanzo Gao, Abigail Bowering, and Wei Zhu). To appear in Inverse Problems and Imaging (2025). arXiv:2412.09079.

•  Neumann problems for p-harmonic functions, and induced nonlocal operators in metric measure spaces. (joint with  Josh Kline, Riikka Korte, Nageswari Shanmugalingam, Marie Snipes), to appear in American Journal of Mathematics  arXiv:2204.00571.

• Variational Solutions to the $\infty$-Poisson equation with respect to H\"ormander vector fields, (joint with Gianmarco Giovannardi, Andrea Pinamonti e Simone Verzellesi),  Mathematische Annalen, 390 no. 2 (2024)  2113-2153.  arXiv:2302.03532

• Regularity Theory of Quasilinear Ellipitic and Parabolic equations in the Heisenberg group. (joint with Giovanna Citti and Xiao Zhong).   Vietnam Jour. of Math. (special number in honor of Carlos Kenig) 52 no. 4 (2024) 807-827.

• Robust Neural Network Approach to System Identification in the High-Noise Regime. (joint with Elisa Negrini and Giovanna Citti),  arXiv:2110.08382  Lecture Notes in Computer Science, (LNCS, volume 14286), Springer Verlag (2023)

• Lipschitz regularity for the parabolic p-Laplacian in the Heisenberg group. (joint with Giovanna Citti and Xiao Zhong).  arXiv:2106.05998 Annales Fennici Mathematici, Vol. 48 No. 2 (2023)

•  System Identification Through Lipschitz Regularized Deep Neural Networks. (joint with Elisa Negrini and Giovanna Citti),  Journal of Computational Physics,  444 (2021), Paper No. 110549, 23 pp arXiv:2009.03288

• Regularity for quasilinear degenerate parabolic equations in the Heisenberg group.  (joint with Giovanna Citti and Nicola Garofalo). Mathematics in Engineering, 2021, 3(1): 1-31, arXiv:2001.08303 

•   Conformal equivalence of visual metrics in pseudo convex domains.  arXiv:1703.00238 (joint with Enrico Le Donne). Mathematische Annalen 379 (2021), no. 1-2, 743–763 

• Analysis on Metric Spaces. (joint with M. Bonk, P. Hajlasz, N. Shanmugalingam, and J. T. Tyson).  Notices Amer. Math. Soc. 67 (2020), no. 2, 253–256.

•  Conformality and Q-harmonicity in Subriemannian manifolds.  arXiv:1603.05548 (joint with Giovanna Citti, Enrico Le Donne and Alessandro Ottazzi).   J. Math. Pures Appl. (9) 122 (2019), 67–124

•   Strong Comparison Principle for $p-$harmonic functions in Carnot-Caratheodory spaces.   (joint with Xiaodan Zhou).  arXiv:1707.06272    Proc. Amer. Math. Soc. 146 (2018), no. 10, 4265–4274(2018).

•   Smoothness of subRiemannian isometries, arXiv:1305.5286 (with Enrico Le Donne). American Journal of Mathematics, 138 (2016), no. 5, 1439-1454.

•  Regularity for subelliptic PDE through uniform estimates in multiscale geometries.  arXiv:1504.04762   (Joint with Giovanna Citti),    Bulletin of Mathematical Sciences, 6 (2016) no. 2, 173-230.

•  Regularity of mean curvature flow of graphs on Lie groups free up to step 2.  arXiv:1502.00874 (joint with Giovanna Citti and Maria Manfredini), Nonlinear Analysis: Theory, Methods and Applications,  126 (2015), 437-450..

•   $L^\infty-$ extremal  mappings in AMLE and Teichmueller theory,. Fully nonlinear PDEs in real and complex geometry and optics, 1–46, Lecture Notes in Math., 2087, Springer, Cham, 2014.

•  Harnack estimates for degenerate parabolic estimates modeled on the subelliptic p-Laplacian. Adv. Math. 257 (2014), 25–65.  arXiv:1306.5650 (with  Benny Avelin, Giovanna Citti and Kaj Nystrom). 

•  Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups.  Anal. Geom. Metr. Spaces 1 (2013), 255–275,  arXiv :1212.6665 (with Giovanna Citti and Maria Manfredini).  

•   Sub-Riemannian Heat Kernels and Mean Curvature of graphs,  Jour. of Functional Analysis, Volume 264, Issue 8, 15 April 2013, Pages 1899–1928,   arXiv:1208.6211 (with Giovanna Citti and Cosimo Senni Guidotti Magnani). (2013)

•  A subelliptic analogue of Aronson-Serrin’s Harnack Inequality. Math. Ann. 357 (2013), no. 3, 1175–1198.arXiv:1109.4596v1 (with Giovanna Citti and Garrett Rea). (2013)

•   An Aronsson type approach to extremal Quasiconformal mappings,   arXiv:1011.2988v1  Jour. Diff. Equations 253, 3  (2012)  851-877. (with Andrew S. Raich).

•  Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups  Trans. Amer. Math. Soc.  362 (2010), 8, 4045-4062 (with S. D. Pauls and J. Tyson)

•   Smoothness  of Lipschitz intrinsic minimal graphs in the Heisenberg group H^n, n>1 ,   Crelle's Journal  (2010) vol. 648, 75-110 (with Giovanna Citti and Maria Manfredini).

•  Generalized mean curvature flow in Carnot groups, Communications in PDE,  34 (2009), 7-9, 937-956 arXiv:0808.3467  (with Giovanna Citti)

• Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$,  Indiana University Mathematical Journal, 58 (2009), 5, 2115-2160.  arXiv:0804.3406 , (with Giovanna Citti and Maria Manfredini).

•  A sub-Riemannian analogue of the mean curvature flow, Oberwolfach Abstract 2008.

•  Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators,   “Perspectives in Partial Differential Equations”, 49-100, Proceedings of Symposia in Pure Mathematics,  79(2008), arXiv:0803.0784, (with Nicola Garofalo and Duy-Minh Nhieu)

•   Isoperimetric inequalities in the Heisenberg group and in the plane.    Lecture Notes of “Seminario Interdisciplinare di Matematica”, Vol. 6 (2007), 93-106

•  On the Mixed boundary value problem for Laplace's operator on Lipschitz planar domains. Math. Annalen, 342 (2008), 1, 91-124  (with R. Brown and L. Lanzani)

•   Ill-posedness of the Cauchy problem for Heisenberg Wave maps  Proc. Amer. Math. Soc.  136 (2008), 5,  1619-1629. (with J. Shatah)

•  A note on the engulfing property and the $\Gamma^{1+ \alpha}$-regularity of convex functions in Carnot Groups,  Proc. Amer. Math. Soc. 134 (2006) 3191-3199. (with D. Maldonado)

•  "The Aronsson-Euler equation for absolutely minimizing lipschitz extensions in Carnot-Caratheodory spaces."  Trans. Amer. Math. Soc. 357 (2005), no. 2, 795--823. (with Thomas Bieske).

•   "Conformality and Q-harmonicity in Carnot groups ", Duke Math. Journal Vol. 135, No. 3 (2006) (with Michael Cowling).

•   "Ahlfors type estimates for the perimeter measure in Carnot-Caratheodory spaces".  Journal of Geometric Analysis, Vol. 16 (3)  (2006)  455-497 . (with Nicola Garofalo).

•   Pointwise Schauder estimates for subelliptic operators in Carnot groups , Proceedings for the AMS-SIAM Harmonic Analysis conference in Mt. Holyhoke, 2001. (with Qing Han).

•  "Examples of Uniform and NTA domains in Carnot groups. " , Proceedings on Analysis and Geometry (Novosibirsk Akademgorodok, 1999) (Russian), 103--121, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000. (with D.M. Nhieu and N. Garofalo).

•   "Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type" , American Journal of Mathematics 124, vol 2, 273--306. (2002) (with D.M. Nhieu and N. Garofalo).

•   A version of a theorem of Dahlberg for the subelliptic Dirichlet problem. Math. Res. Lett. 5 (1998), no. 4, 541--549 (DM Nhieu and N. Garofalo).

•   Legendrian energy minimizers (with Fang-Hua Lin) Calc. Var. Partial Differential Equations 12 (2001), no. 2, 145--171.

•   Regularity for quasilinear equations and $1-$quasiconformal maps in Carnot groups, Math. Annalen 313 (1999), no. 2, 263--295.

•  "Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of H\"ormander type ", Journal of the European Mathematical Society, 2 (2003). (with N. Garofalo).

•   " Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics.", ,, J. Fourier Anal. Appl. 4 (1998), no. 4-5, 403--432. (with N. Garofalo).

•   Regularity for quasilinear equations in the Heisenberg group, Communications in Pure and Applied Mathematics vol. L, (1997), pages 867--889.

•  Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, American Journal of Mathematics, vol. 118, no. 6 (1996), pages 1153-1196 (with D. Danielli and N. Garofalo).

•  Uniform domains and quasiconformal mappings on the Heisenberg group , Man. Math., 86 (1995), pages 267-281 (with P. Tang).

•   Subelliptic mollifiers and a basic pointwise estimate of Poincar\'e type, Math. Z. (1997) (with D. Danielli and N. Garofalo).

•  The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. in Analysis and Geometry 2(2) (1994), pages 203--215 (with D. Danielli and N. Garofalo).

•  Subelliptic mollifiers and a characterization of Rellich and Poincar\'e domains, Rend. Sem. Mat. Univ. Pol. Torino, 51 (4) (1993) pages 361-386 (with D. Danielli and N. Garofalo).

•  An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Diff. Eq., 18 (1993), pages 1765-1794 (with D. Danielli and N. Garofalo).