University of Illinois at Urbana-Champaign


Research supported by NSF Graduate and Postdoctoral Research Fellowships and NSF grants DMS-0228807, DMS-0555869, DMS-0901620, DMS-1201875, and DMS-1600650, and Simons Collaboration Grant 353627.

Any opinions, findings, conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or the Simons Foundation.

Books and Monographs

  1. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem with L. Capogna, D. Danielli, and S.D. Pauls, Progress in Mathematics, vol. 259, Birkhauser, 2007.

  2. Conformal dimension: theory and application with J. M. Mackay, University Lecture Series, vol. 54, American Mathematical Society, 2010.

  3. Sobolev spaces on metric measure spaces: an approach based on upper gradients with J. Heinonen, P. Koskela and N. Shanmugalingam, New Mathematical Monographs, vol. 27, Cambridge University Press, 2015.


  1. PDF    Heat content and horizontal mean curvature in the Heisenberg group, with Jing Wang, preprint, 2017.

  2. PDF    Superposition of fundamental solutions of second order quasilinear equations, preprint, 2016.

  3. PDF    Conformal graph directed Markov systems on Carnot groups, with V. Chousionis and M. Urbański, to appear in Memoirs of the AMS.   

  4. PDF    Quasiconvexity in the Heisenberg group, with D. Herron and A. Lukyanenko, to appear in Geom. Dedicata.

  5. PDF    Heisenberg quasiregular ellipticity, with K. Fässler and A. Lukyanenko, to appear in Rev. Mat. Iberoamericana.

  6. PDF    A new proof of the C regularity of C2 conformal mappings on the Heisenberg group, with A. D. Austin, to appear in Colloq. Math.

  7. PDF    Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the sub-Riemannian Heisenberg group, with Z. M. Balogh and E. Vecchi, to appear in Math. Z.   
  8. PDF    Frequency of Sobolev dimension distortion of horizontal subgroups of Heisenberg groups, with Z. M. Balogh and K. Wildrick, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5).    Abstract

  9. PDF    Quasiconformal mappings that highly distort dimensions of many parallel lines, with Z. M. Balogh and K. Wildrick, Ann. Acad. Sci. Fenn. Ser. A I Math., 42 (2017) 61-72.    Abstract

  10. PDF    Removable sets for homogeneous linear PDE in Carnot groups, with V. Chousionis, J. Analyse Math., 128 no. 1 (2016) 215-238.    Abstract

  11. PDF    Marstrand's density theorem in the Heisenberg group, with V. Chousionis, Bull. London Math. Soc., 47 no. 5 (2015) 771-788.    Abstract

  12. PDF    Removable sets for Lipschitz harmonic functions on Carnot groups, with V. Chousionis and V. Magnani, Calc. Var. PDE, 53 no. 3-4 (2015) 755-780.    Abstract

  13. PDF    On transverse submanifolds and their measure, with V. Magnani and D. Vittone, J. Analyse Math., 125 (2015), 319-351.    Abstract

  14. PDF    Homotopy groups of spheres and Lipschitz homotopy groups of the Heisenberg groups, with P. Hajlasz and A. Schikorra, Geom. Funct. Analysis, 24 (2014) 245-268.    Abstract

  15. PDF    On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg group target, with N. DeJarnette, P. Hajlasz and A. Lukyanenko, Conf. Geom. Dynam., 18 (2014), 119-156.    Abstract

  16. PDF    Grassmannian frequency of Sobolev dimension distortion, with Z. M. Balogh and P. Mattila, Comput. Methods Funct. Theory, 14 no. 2-3 (2014) 505-523.    Abstract

  17. PDF    Dimension distortion by Sobolev mappings in foliated metric spaces, with Z. M. Balogh and K. Wildrick, Analysis and Geometry in MetricSpaces, 1 (2013) 232-254.    Abstract

  18. PDF    Modulus and Poincaré inequalities on non-self-similar Sierpiński carpets, with J. M. Mackay and K. Wildrick, Geom. Funct. Analysis, 23 no. 3 (2013) 985-1034.    Abstract

  19. PDF    Quasiregular maps and the conductivity equation in the Heisenberg group, with A. Isopoussu and K. Peltonen, In the Tradition of Ahlfors-Bers, VI AMS Contemp. Math. 590 (2013), 61-75.    Abstract

  20. PDF    The effect of projections on dimension in the Heisenberg group with Z. M. Balogh, E. Durand Cartagena, K. Fässler, and P. Mattila, Rev. Mat. Iberoamericana, 29 no. 2 (2013), 382-432.    Abstract

  21. PDF    Frequency of Sobolev and quasiconformal dimension distortion, with Z. M. Balogh and R. Monti, J. Math. Pures Appl., 99 no. 2 (2013), 125–149.    Abstract

  22. PDF    Projection and slicing theorems in Heisenberg groups with Z. M. Balogh, K. Fässler, and P. Mattila, Advances in Mathematics, 231 no. 2 (2012), 569-604.    Abstract

  23. PDF    Riesz potentials and p-superharmonic functions in Lie groups of Heisenberg type, with N. Garofalo, Bulletin London Math. Soc., 44 no. 2 (2012) 353-366.    Abstract

  24. PDF    Rectifiable curves in Sierpiński carpets, with E. Durand Cartagena, Indiana Univ. Math. J. 60 (2011) 285-310.    Abstract, see also the erratum

  25. PDF    An Invitation to Cauchy-Riemann and Sub-Riemannian Geometries, with J.P. D'Angelo, Notices Amer. Math. Soc. 57 no. 2 (2010) 208-219.    Abstract

  26. PDF    Exceptional sets for self-similar fractals in Carnot groups with Z. M. Balogh, R. Berger and R. Monti, Math. Proc. Cambridge Philos. Soc. 149 (2010) no. 1, 147-172.    Abstract

  27. PDF    Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups, with L. Capogna and S.D. Pauls, Trans. Amer. Math. Soc. 362 (2010) 4045-4062.    Abstract

  28. PDF    Helical CR structures and sub-Riemannian geodesics, with J.P. D'Angelo, Complex Var. Elliptic Equ. 54 (2009) 205-221.    Abstract

  29. PDF    Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry in Carnot groups, with Z.M. Balogh and B. Warhurst, Advances in Mathematics 220 (2009) 560-619.    Abstract

  30. PDF    Sobolev Peano cubes, with P. Hajlasz, Michigan Math. J. 56 (2008) 687-702.    Abstract

  31. PDF    Gromov's dimension comparison problem on Carnot groups, with Z.M. Balogh and B. Warhurst, C. R. Acad. Sci. Paris Ser. I, Math 346 (2008) 135-138.    Abstract

  32. PDF    Global conformal Assouad dimension in the Heisenberg group, Conf. Geom. Dynam. 12 (2008) 32-57.    Abstract

  33. PS    PDF    Hyperbolic and quasisymmetric structure of hyperspaces, with L.V. Kovalev, In the Tradition of Ahlfors-Bers, IV , AMS Contemp. Math. 432 (2007), 151-166.    Abstract

  34. PDF    Sharp weighted Young's inequalities and Moser-Trudinger inequalities on groups of Heisenberg type and Grushin spaces , Potential Anal. 24 (2006) 357-384.    Abstract

  35. PDF    Quasiconformal dimensions of self-similar sets, with J.-M. Wu, Rev. Mat. Iberoamericana 22 (2006) 205-258.    Abstract

  36. PDF    Lifts of Lipschitz maps and horizontal fractals on the Heisenberg group, with Z.M. Balogh and R. Hofer-Isenegger, Erg. Theory. and Dynam. Systems 26 (2006) 621-651.    Abstract

  37. PDF    Bi-Lipschitz embeddings of hyperspaces of compact sets Fund. Math. 187 (2005) 229-254.    Abstract

  38. PS    PDF    Smooth quasiregular maps with branching in R^n, with R.P. Kaufman and J.-M. Wu, Publ. Math IHES. 101 (2005) 209-241.    Abstract

  39. PS/PDF/DVI    Characterizations of snowflake metric spaces, with J.-M. Wu, Ann. Acad. Sci. Fenn. Ser. A I, Math. 30 (2005) 313-336.    Abstract

  40. PDF    Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group, with Z.M. Balogh, Proc. London Math. Soc. 91 (2005) 153-183.    Abstract

  41. PDF    Counterexamples to Tischler's strong form of Smale's Mean Value Conjecture Bull. London Math. Soc. 37 (2005) 95-106.    Abstract

  42. PDF    Dirichlet forms, Poincaré inequalities and the Sobolev spaces of Korevaar-Schoen, with P. Koskela and N. Shanmugalingam, Pot. Anal. 21 (2004) 241-262.    Abstract

  43. PS    Quasiconformal maps on metric spaces: questions and conjectures, in Future Trends in Geometric Function Theory: Rolf Nevanlinna Colloquium Workshop, Jyväskylä, 2003 (D. Herron, ed.), Report. Univ. Jyväskylä 92 (2003) 249-262.    Abstract

  44. PDF    Fundamental solution for the Q-Laplacian and sharp Moser-Trudinger inequality in Carnot groups, with Z.M. Balogh and J.J. Manfredi, J. Funct. Anal. 204 (2003) 35-49.    Abstract

  45. PS    PDF    Potential theory in Carnot groups, with Z.M. Balogh, Harmonic Analysis at Mount Holyoke (W. Beckner, A. Nagel, A. Seeger and H. F. Smith, eds.), AMS Contemp. Math. 320 (2003) 15-27.    Abstract

  46. PDF    Polar coordinates in Carnot groups, with Z.M. Balogh, Math. Z. 241 (2002) 697-730.    Abstract

  47. PDF    Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups, with Z.M. Balogh and I. Holopainen, Math. Ann. 324 (2002) 159-186.    Abstract

  48. PDF    Quasihyperbolic boundary conditions and Poincaré domains, with P. Koskela and J. Onninen, Math. Ann. 323 (2002) 811-830.    Abstract

  49. PDF    Quasihyperbolic boundary conditions and capacity: Hölder continuity of quasiconformal mappings, with P. Koskela and J. Onninen, Comm. Math. Helv. 76 (2001) 416-435.    Abstract

  50. PS/PDF    Lowering the Assouad dimension by quasiconformal mappings, Illinois J. Math. 45 (2001) 641-656.    Abstract

  51. PS/PDF/DVI    Locally minimal sets for conformal dimension, with C.J. Bishop, Ann. Acad. Sci. Fenn. Ser. A I Math. 26 (2001) 361-373.    Abstract

  52. PS/PDF/DVI    Conformal dimension of the antenna set, with C.J. Bishop, Proc. Amer. Math. Soc. 129 (2001) 3631-3636.    Abstract

  53. PS    On the conformal Martin boundary of domains in metric spaces, with I. Holopainen and N. Shanmugalingam, Papers on Analysis: A volume dedicated to Olli Martio on the occasion of his 60th birthday (J. Heinonen, T. Kilpeläinen, and P. Koskela, eds.), Report. Univ. Jyväskylä 83 (2001) 147-168.    Abstract

  54. PS    PDF    Sobolev classes of Banach space-valued functions and quasiconformal mappings, with J. Heinonen, P. Koskela and N. Shanmugalingam, J. Analyse Math. 85 (2001) 87-139.    Abstract

  55. PS/PDF/DVI    Metric and geometric quasiconformality in Ahlfors regular Loewner spaces, Conf. Geom. Dynam. 5 (2001) 21-73.    Abstract

  56. PDF    Analytic properties of locally quasisymmetric mappings from Euclidean domains, Indiana Univ. Math. J. 49 (2000) 995-1016.    Abstract

  57. PS/PDF/DVI    Sets of minimal Hausdorff dimension for quasiconformal mappings, Proc. Amer. Math. Soc. 128 (2000) 3361-3367.    Abstract

  58. PS/PDF/DVI    Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998) 525-548.    Abstract

Lecture Notes

  1. Lecture notes    Notes on a theorem of Jeff Cheeger, lecture notes from a course given by Juha Heinonen at the University of Michigan in Spring 1999.

Department of Mathematics
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University of Illinois at Urbana-Champaign
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