Erika Battaglia[1]; Stefano Biagi[1]; Andrea Bonfiglioli[1]

  • [1] Dipartimento di Matematica, Università di Bologna Piazza di Porta San Donato, 5 40126 Bologna (Italy)

Annales de l’institut Fourier (0)

  • Volume: 0, Issue: 0, page 1-43
  • ISSN: 0373-0956

How to cite


Battaglia, Erika, Biagi, Stefano, and Bonfiglioli, Andrea. "null." Annales de l’institut Fourier 0.0 (0): 1-43. <http://eudml.org/doc/275360>.

affiliation = {Dipartimento di Matematica, Università di Bologna Piazza di Porta San Donato, 5 40126 Bologna (Italy); Dipartimento di Matematica, Università di Bologna Piazza di Porta San Donato, 5 40126 Bologna (Italy); Dipartimento di Matematica, Università di Bologna Piazza di Porta San Donato, 5 40126 Bologna (Italy)},
author = {Battaglia, Erika, Biagi, Stefano, Bonfiglioli, Andrea},
journal = {Annales de l’institut Fourier},
language = {eng},
number = {0},
pages = {1-43},
publisher = {Association des Annales de l’institut Fourier},
url = {http://eudml.org/doc/275360},
volume = {0},
year = {0},

AU - Battaglia, Erika
AU - Biagi, Stefano
AU - Bonfiglioli, Andrea
JO - Annales de l’institut Fourier
PY - 0
PB - Association des Annales de l’institut Fourier
VL - 0
IS - 0
SP - 1
EP - 43
LA - eng
UR - http://eudml.org/doc/275360
ER -


  1. Beatrice Abbondanza, Andrea Bonfiglioli, The Dirichlet problem and the inverse mean-value theorem for a class of divergence form operators, J. Lond. Math. Soc. (2) 87 (2013), 321-346 Zbl1266.31004
  2. H. Aimar, L. Forzani, R. Toledano, Hölder regularity of solutions of PDE’s: a geometrical view, Comm. Partial Differential Equations 26 (2001), 1145-1173 Zbl1017.35026
  3. Kazuo Amano, A necessary condition for hypoellipticity of degenerate elliptic-parabolic operators, Tokyo J. Math. 2 (1979), 111-120 Zbl0429.35026
  4. Martin T. Barlow, Richard F. Bass, Stability of parabolic Harnack inequalities, Trans. Amer. Math. Soc. 356 (2004), 1501-1533 (electronic) Zbl1034.60070
  5. Erika Battaglia, Andrea Bonfiglioli, Normal families of functions for subelliptic operators and the theorems of Montel and Koebe, J. Math. Anal. Appl. 409 (2014), 1-12 Zbl1325.35011
  6. Denis R. Bell, Salah Eldin A. Mohammed, An extension of Hörmander’s theorem for infinitely degenerate second-order operators, Duke Math. J. 78 (1995), 453-475 Zbl0840.60053
  7. A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, (2007), Springer, Berlin Zbl1128.43001
  8. Andrea Bonfiglioli, Ermanno Lanconelli, Subharmonic functions in sub-Riemannian settings, J. Eur. Math. Soc. (JEMS) 15 (2013), 387-441 Zbl1270.31002
  9. Andrea Bonfiglioli, Ermanno Lanconelli, Andrea Tommasoli, Convexity of average operators for subsolutions to subelliptic equations, Anal. PDE 7 (2014), 345-373 Zbl1302.35133
  10. Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), 277-304 xii Zbl0176.09703
  11. Marco Bramanti, Luca Brandolini, Ermanno Lanconelli, Francesco Uguzzoni, Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities, Mem. Amer. Math. Soc. 204 (2010) Zbl1218.35001
  12. Marcel Brelot, Axiomatique des fonctions harmoniques, 1965 (1969), Les Presses de l’Université de Montréal, Montreal, Que. Zbl0084.31401
  13. Sagun Chanillo, Richard L. Wheeden, Harnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986), 1111-1134 Zbl0634.35035
  14. Michael Christ, Hypoellipticity in the infinitely degenerate regime, Complex analysis and geometry (Columbus, OH, 1999) 9 (2001), 59-84, de Gruyter, Berlin Zbl1015.32032
  15. Giovanna Citti, Nicola Garofalo, Ermanno Lanconelli, Harnack’s inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), 699-734 Zbl0795.35018
  16. C. Constantinescu, A. Cornea, On the axiomatic of harmonic functions. I, Ann. Inst. Fourier (Grenoble) 13 (1963), 373-388 Zbl0122.34001
  17. Virginia De Cicco, Maria Agostina Vivaldi, Harnack inequalities for Fuchsian type weighted elliptic equations, Comm. Partial Differential Equations 21 (1996), 1321-1347 Zbl0859.35013
  18. Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25-43 Zbl0084.31901
  19. Giuseppe Di Fazio, Cristian E. Gutiérrez, Ermanno Lanconelli, Covering theorems, inequalities on metric spaces and applications to PDE’s, Math. Ann. 341 (2008), 255-291 Zbl1149.46029
  20. J. Dieudonné, Éléments d’analyse. Tome VII. Chapitre XXIII. Première partie, (1978), Gauthier-Villars, Paris Zbl0402.58011
  21. E. Fabes, D. Jerison, C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), vi, 151-182 Zbl0488.35034
  22. E. B. Fabes, C. E. Kenig, D. Jerison, Boundary behavior of solutions to degenerate elliptic equations, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) (1983), 577-589, Wadsworth, Belmont, CA 
  23. Eugene B. Fabes, Carlos E. Kenig, Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77-116 Zbl0498.35042
  24. V. S. Fediĭ, A certain criterion for hypoellipticity, Mat. Sb. (N.S.) 85 (127) (1971), 18-48 
  25. C. Fefferman, D. H. Phong, The uncertainty principle and sharp Gȧrding inequalities, Comm. Pure Appl. Math. 34 (1981), 285-331 Zbl0458.35099
  26. C. Fefferman, D. H. Phong, Subelliptic eigenvalue problems, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) (1983), 590-606, Wadsworth, Belmont, CA Zbl0503.35071
  27. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207 Zbl0312.35026
  28. G. B. Folland, J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, (1972), Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo Zbl0247.35093
  29. G. B. Folland, E. M. Stein, Estimates for the ¯ b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522 Zbl0293.35012
  30. Bruno Franchi, Ermanno Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations 9 (1984), 1237-1264 Zbl0589.46023
  31. Bruno Franchi, Ermanno Lanconelli, Une condition géométrique pour l’inégalité de Harnack, J. Math. Pures Appl. (9) 64 (1985), 237-256 Zbl0599.35134
  32. Nicola Garofalo, Ermanno Lanconelli, Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients, Math. Ann. 283 (1989), 211-239 Zbl0638.35003
  33. Nicola Garofalo, Ermanno Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc. 321 (1990), 775-792 Zbl0719.35007
  34. Alexander Grigor’yan, Laurent Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble) 55 (2005), 825-890 Zbl1115.58024
  35. Cristian E. Gutiérrez, Harnack’s inequality for degenerate Schrödinger operators, Trans. Amer. Math. Soc. 312 (1989), 403-419 Zbl0685.35020
  36. Cristian E. Gutiérrez, Ermanno Lanconelli, Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for X -elliptic operators, Comm. Partial Differential Equations 28 (2003), 1833-1862 Zbl1064.35036
  37. W. Hebisch, L. Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001), 1437-1481 Zbl0988.58007
  38. Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171 Zbl0156.10701
  39. Lars Hörmander, The analysis of linear partial differential operators. I, 256 (1990), Springer-Verlag, Berlin Zbl0712.35001
  40. Sapto Indratno, Diego Maldonado, Sharad Silwal, On the axiomatic approach to Harnack’s inequality in doubling quasi-metric spaces, J. Differential Equations 254 (2013), 3369-3394 Zbl1270.35230
  41. David Jerison, Antonio Sánchez-Calle, Subelliptic, second order differential operators, Complex analysis, III (College Park, Md., 1985–86) 1277 (1987), 46-77, Springer, Berlin Zbl0634.35017
  42. Velimir Jurdjevic, Geometric control theory, 52 (1997), Cambridge University Press, Cambridge Zbl0940.93005
  43. Juha Kinnunen, Niko Marola, Michele Miranda, Fabio Paronetto, Harnack’s inequality for parabolic De Giorgi classes in metric spaces, Adv. Differential Equations 17 (2012), 801-832 Zbl1255.30057
  44. Alessia E. Kogoj, A control condition for a weak Harnack inequality, Nonlinear Anal. 75 (2012), 4198-4204 Zbl1243.35045
  45. J. J. Kohn, Boundaries of complex manifolds, Proc. Conf. Complex Analysis (Minneapolis, 1964) (1965), 81-94, Springer, Berlin Zbl0166.36003
  46. J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal. 159 (1998), 203-216 Zbl0937.35024
  47. J. J. Kohn, L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443-492 Zbl0125.33302
  48. S. Kusuoka, D. Stroock, Applications of the Malliavin calculus. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), 1-76 Zbl0568.60059
  49. Peter A. Loeb, Bertram Walsh, The equivalence of Harnack’s principle and Harnack’s inequality in the axiomatic system of Brelot, Ann. Inst. Fourier (Grenoble) 15 (1965), 597-600 Zbl0132.33802
  50. Julián López-Gómez, The strong maximum principle, Mathematical analysis on the self-organization and self-similarity (2009), 113-123, Res. Inst. Math. Sci. (RIMS), Kyoto Zbl1193.35029
  51. Guozhen Lu, On Harnack’s inequality for a class of strongly degenerate Schrödinger operators formed by vector fields, Differential Integral Equations 7 (1994), 73-100 Zbl0827.35032
  52. Ahmed Mohammed, Harnack’s inequality for solutions of some degenerate elliptic equations, Rev. Mat. Iberoamericana 18 (2002), 325-354 Zbl1140.35364
  53. P. Montel, Leçons sur les familles normales de fonctions analytiques et leurs applications. Recueillies et rédigées par J. Barbotte., VIII + 306 p. Paris, Gauthier-Villars (1927). (1927) Zbl53.0303.02
  54. Yoshinori Morimoto, A criterion for hypoellipticity of second order differential operators, Osaka J. Math. 24 (1987), 651-675 Zbl0644.35023
  55. Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591 Zbl0111.09302
  56. John Nash, Parabolic equations, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 754-758 Zbl0078.08704
  57. Alberto Parmeggiani, A remark on the stability of C -hypoellipticity under lower-order perturbations, J. Pseudo-Differ. Oper. Appl. 6 (2015), 227-235 Zbl1317.47053
  58. Andrea Pascucci, Sergio Polidoro, A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations, J. Math. Anal. Appl. 282 (2003), 396-409 Zbl1026.35056
  59. Andrea Pascucci, Sergio Polidoro, Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators, Trans. Amer. Math. Soc. 358 (2006), 4873-4893 (electronic) Zbl1172.35339
  60. Patrizia Pucci, James Serrin, The maximum principle, (2007), Birkhäuser Verlag, Basel Zbl1134.35001
  61. Linda Preiss Rothschild, E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320 Zbl0346.35030
  62. L. Saloff-Coste, Parabolic Harnack inequality for divergence-form second-order differential operators, Potential Anal. 4 (1995), 429-467 Zbl0840.31006
  63. James Serrin, On the Harnack inequality for linear elliptic equations, J. Analyse Math. 4 (1955/56), 292-308 Zbl0070.32302
  64. E. M. Stein, An example on the Heisenberg group related to the Lewy operator, Invent. Math. 69 (1982), 209-216 Zbl0515.58032
  65. François Trèves, Topological vector spaces, distributions and kernels, (1967), Academic Press, New York-London Zbl0171.10402
  66. Pietro Zamboni, Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions, J. Differential Equations 182 (2002), 121-140 Zbl1014.35036

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