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

•  Existence and Uniqueness of Green functions for the Cheeger p-Laplacian in PI spaces. (joint with M. Bonk and X. Zhou).  Preprint ArXiv:2211.11974

•  Neumann problems for p-harmonic functions, and induced nonlocal operators in metric measure spaces. (joint with  Josh Kline, Riikka Korte, Nageswari Shanmugalingam, Marie Snipes), under review 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),  to appear in Mathematische Annalen.  arXiv:2302.03532

• Regularity Theory of Quasilinear Ellipitic and Parabolic equations in the Heisenberg group. (joint with G. Citti and X. Zhong).   Vietnam Jour. of Math.  (2023) 

• Robust Neural Network Approach to System Identification in the High-Noise Regime. (joint with E. Negrini and G. 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 G. Citti and X. Zhong).  arXiv:2106.05998 Annales Fennici Mathematici, Vol. 48 No. 2 (2023)

•  System Identification Through Lipschitz Regularized Deep Neural Networks. (joint with E. Negrini and G. 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 G. Citti and N. 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 G. 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 G. Citti and M. 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).