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Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

  • Andrea BonfiglioliEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 275)

Abstract

We outline several results of Potential Theory for a class of linear partial differential operators \(\mathcal {L}\) of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for \(\mathcal {L}\); under different geometrical assumptions on \(\mathcal {L}\) (mainly, under global doubling/Poincaré assumptions), it is described how to obtain an invariant, non-homogeneous Harnack inequality. When \(\mathcal {L}\) is equipped with a global fundamental solution \(\varGamma \), further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on \(\mathcal {L}\) ensuring that such a \(\varGamma \) exists.

Keywords

Fundamental solutions Hypoelliptic operators Harnack inequalities Potential theory 

Notes

Acknowledgements

The results of this paper were presented by the author at the Conference “Noncommutative Analysis and Partial Differential Equations”, 11–15 April, 2016, Imperial College, London; the author wishes to express his gratitude to the Organizing Committee of the Conference for the hospitality.

References

  1. 1.
    Abbondanza, B., Bonfiglioli, A.: On the Dirichlet problem and the inverse mean value theorem for a class of divergence form operators. J. Lond. Math. Soc. 1–26 (2012).  https://doi.org/10.1112/jlms/jds050. MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agrachev, A., Boscain, U., Gauthier, J.-P., Rossi, F.: The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256, 2621–2655 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aimar, H., Forzani, L., Toledano, R.: Hölder regularity of solutions of PDE’s: a geometrical view. Commun. Partial Differ. Equ. 26, 1145–1173 (2001)CrossRefGoogle Scholar
  4. 4.
    Amano, K.: A necessary condition for hypoellipticity of degenerate elliptic-parabolic operators. Tokyo J. Math. 2, 111–120 (1979)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barlow, M.T., Bass, R.F.: Stability of parabolic Harnack inequalities. Trans. Am. Math. Soc. 356, 1501–1533 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Battaglia, E., Biagi, S., Bonfiglioli, A.: The strong maximum principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators. Ann. Inst. Fourier (Grenoble) 66(2), 589–631 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Battaglia, E., Bonfiglioli, A.: Normal families of functions for subelliptic operators and the theorems of Montel and Koebe. J. Math. Anal. Appl. 409, 1–12 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Battaglia, E., Bonfiglioli, A.: An invariant Harnack inequality for a class of subelliptic operators under global doubling and Poincaré assumptions, and applications. J. Math. Anal. Appl. 460, 302–320 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bauer, W., Furutani, K., Iwasaki, C.: Fundamental solution of a higher step Grushin type operator. Adv. Math. 271, 188–234 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Beals, R.: Exact fundamental solutions, “Equations aux D‘eriv‘ees Partielles” (Saint-Jean-de-Monts, 1998). University of Nantes, Nantes. Exp. No. I, pp. 9 (1998)Google Scholar
  11. 11.
    Beals, R., Gaveau, B., Greiner, P.: On a geometric formula for the fundamental solution of subelliptic Laplacians. Math. Nachr. 181, 81–163 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Beals, R., Gaveau, B., Greiner, P.: The green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy Riemann complexes. Adv. Math. 121(2), 288–345 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Beals, R., Gaveau, B., Greiner, P., Kannai, Y.: Exact fundamental solutions for a class of degenerate elliptic operators. Commun. Partial Differ. Equ. 24(3–4), 719–742 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Beals, R., Gaveau, B., Greiner, P., Kannai, Y.: Transversally elliptic operators. Bull. Sci. Math. 128(7), 531–576 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Beals, R., Greiner, P., Gaveau, B.: Fundamental solutions for some highly degenerate elliptic operators. J. Funct. Anal. 165(2), 407–429 (1999)Google Scholar
  16. 16.
    Biagi, S., Bonfiglioli, A.: The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method. Proc. Lond. Math. Soc. 114, 855–889 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Biagi, S., Bonfiglioli, A.: Global heat kernels for homogeneous Hörmander operators and applications, in preparation (2017)Google Scholar
  18. 18.
    Bieske, T.: Fundamental solutions to the \(p\)-Laplace equation in a class of Grushin vector fields. Electron. J. Differ. Equ. 84, 1–10 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bieske, T., Gong, J.: The \(p\)-Laplace equation on a class of Grushin-type spaces. Proc. Am. Math. Soc. 134, 3585–3594 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Biroli, M., Mosco, U.: Sobolev inequalities on homogeneous spaces. Potential theory and degenerate partial differential operators (Parma). Potential Anal. 4, 311–324 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Bonfiglioli, A., Lanconelli, E.: Liouville-type theorems for real sub-Laplacians. Manuscr. Math. 105, 111–124 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Bonfiglioli, A., Lanconelli, E.: Subharmonic functions on Carnot groups. Math. Ann. 325, 97–122 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bonfiglioli, A., Lanconelli, E.: Gauge functions, Eikonal equations and Bôcher’s theorem on stratified Lie groups. Calc. Var. Partial Differ. Equ. 30, 277–291 (2007)CrossRefGoogle Scholar
  24. 24.
    Bonfiglioli, A., Lanconelli, E.: Subharmonic functions in sub-Riemannian settings. J. Eur. Math. Soc. 15, 387–441 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Bonfiglioli, A., Lanconelli, E., Tommasoli, A.: Convexity of average operators for subsolutions to subelliptic equations. Anal. PDE 7, 345–373 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)zbMATHGoogle Scholar
  27. 27.
    Bony, J.-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19, 277–304 (1969)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Boscain, U., Gauthier, J.-P., Rossi, F.: The hypoelliptic heat kernel over three-step nilpotent Lie groups. Sovrem. Mat. Fundam. Napravl. 42, 48–61 (2011) (Transl. in: J. Math. Sci. (N.Y.) 199(6), 614–628, 2014)Google Scholar
  29. 29.
    Bramanti, M., Brandolini, L., Lanconelli, E., Uguzzoni, F.: Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities. Mem. Am. Math. Soc. 204(961) (2010)CrossRefGoogle Scholar
  30. 30.
    Brelot, M.: Axiomatique des fonctions harmoniques, Séminaire de Mathématiques Supérieures, 14 (té, 1965). Les Presses de l’Université de Montréal, Montréal (1969)Google Scholar
  31. 31.
    Calin, O., Chang, D.-C., Furutani, K., Iwasaki, C.: Heat kernels for elliptic and sub-elliptic operators. Methods and techniques. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York (2011)CrossRefGoogle Scholar
  32. 32.
    Capogna, L., Danielli, D., Garofalo, N.: Capacitary estimates and subelliptic equations. Am. J. Math. 118, 1153–1196 (1996)CrossRefGoogle Scholar
  33. 33.
    Chanillo, S., Wheeden, R.L.: Harnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 11, 1111–1134 (1986)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Christ, M.: Hypoellipticity in the infinitely degenerate regime. In: J.D. McNeal (ed.) Complex Analysis and Geometry, Ohio State University Mathematical Research Institute Publication, vol. 9, pp. 59–84. Walter de Gruyter, Berlin (2001)zbMATHGoogle Scholar
  35. 35.
    Citti, G., Garofalo, N., Lanconelli, E.: Harnack’s inequality for sum of squares of vector fields plus a potential. Am. J. Math. 115, 699–734 (1993)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Di Fazio, G., Gutiérrez, C.E., Lanconelli, E.: Covering theorems, inequalities on metric spaces and applications to PDE’s. Math. Ann. 341, 255–291 (2008)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Fabes, E.B., Jerison, D., Kenig, C.E.: The Wiener test for degenerate elliptic equations. Ann. Inst. Fourier (Grenoble) 32, 151–182 (1982)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7, 77–116 (1982)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Fefferman, C., Phong, D.H.: Subelliptic eigenvalue problems, Wadsworth Math. Ser. (Chicago, III., 1981), Wadsworth, Belmont, CA, pp. 590–606 (1983)Google Scholar
  40. 40.
    Fefferman, C., Phong, D.H.: The uncertainty principle and sharp Garding inequalities. Commun. Pure Appl. Math. 34, 285–331 (1981)CrossRefGoogle Scholar
  41. 41.
    Fedi\(\breve{\rm \i }\), V.S.: On a criterion for hypoellipticity. Math. USSR Sb. 14, 15–45 (1971)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Folland, G.B.: Subelliptic estimates and function spaces on nilpotent lie groups. Ark. MAt. 13, 161–207 (1975)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Folland, G.B.: On the Rothschild-Stein lifting theorem. Commun. Partial Differ. Equ. 2, 161–207 (1977)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Franchi, B., Hajłasz, P., Koskela, P.: Definitions of Sobolev classes on metric spaces. Ann. Inst. Fourier (Grenoble) 49, 1903–1924 (1999)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Franchi, B., Lanconelli, E.: Une condition géométrique pour l’inégalité de Harnack. J. Math. Pures Appl. 64, 237–256 (1985)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Franchi, B., Lu, G., Wheeden, R.L.: A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type. Int. Math. Res. Not. 1, 1–14 (1996)CrossRefGoogle Scholar
  47. 47.
    Franchi, B., Serapioni, R., Serra Cassano, F.: Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll. U.M.I B (7) 11, 83–117 (1997)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Garofalo, N., Lanconelli, E.: Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients. Math. Ann. 283, 211–239 (1989)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Garofalo, N., Lanconelli, E.: Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type. Trans. Am. Math. Soc. 321, 775–792 (1990)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Garofalo, N., Nhieu, D.-M.: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)CrossRefGoogle Scholar
  51. 51.
    Grigor’yan, A., Saloff-Coste, L.: Stability results for Harnack inequalities. Ann. Inst. Fourier (Grenoble) 55, 825–890 (2005)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Gutiérrez, C.E.: Harnack’s inequality for degenerate Schrödinger operators. Trans. Am. Math. Soc. 312, 403–419 (1989)CrossRefGoogle Scholar
  53. 53.
    Gutiérrez, C.E., Lanconelli, E.: Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for X-elliptic operators. Commun. Partial Differ. Equ. 28, 1833–1862 (2003)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51, 1437–1481 (2001)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145 (2000)CrossRefGoogle Scholar
  56. 56.
    Heinonen, J., Holopainen, I.: Quasiregular maps on Carnot groups. J. Geom. Anal. 7, 109–148 (1997)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Indratno, S., Maldonado, D., Silwal, S.: On the axiomatic approach to Harnack’s inequality in doubling quasi-metric spaces. J. Differ. Equ. 254, 3369–3394 (2013)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Jerison, D.S., Sánchez-Calle, A.: Subelliptic, second order differential operators. In: Complex Analysis III, Proceeding Spec. Year, College Park 1985–1986. Lecture Notes Mathematics, vol. 1277, pp. 46–77 (1987)Google Scholar
  59. 59.
    Kinnunen, J., Marola, N., Miranda, M., Paronetto, F.: Harnack’s inequality for parabolic De Giorgi classes in metric spaces. Adv. Differ. Equ. 17, 801–832 (2012)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Kogoj, A.E., Lanconelli, E.: \(X\)-elliptic operators and \(X\)-elliptic distances, Contributions in honor of the memory of Ennio De Giorgi. Ricerche Mat. 49, 223–243 (2000)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Kogoj, A.E., Lanconelli, E.: Liouville theorem for \(X\)-elliptic operators. Nonlinear Anal. 70, 2974–2985 (2009)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Kohn, J.J.: Hypoellipticity of some degenerate subelliptic operators. J. Funct. Anal. 159, 203–216 (1998)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32, 1–76 (1985)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Mohammed, A.: Harnack’s inequality for solutions of some degenerate elliptic equations. Rev. Mat. Iberoamericana 18, 325–354 (2002)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Morimoto, Y.: A criterion for hypoellipticity of second order differential operators. Osaka J. Math. 24, 651–675 (1987)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Parmeggiani, A.: A remark on the stability of \(C^\infty \)-hypoellipticity under lower-order perturbations. J. Pseudo-Differ. Oper. Appl. 6, 227–235 (2015)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Pascucci, A., Polidoro, S.: A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations. J. Math. Anal. Appl. 282, 396–409 (2003)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Pascucci, A., Polidoro, S.: Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators. Trans. Am. Math. Soc. 358, 4873–4893 (2006)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137, 247–320 (1976)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Saloff-Coste, L.: Parabolic Harnack inequality for divergence-form second-order differential operators. Potential Anal. 4, 429–467 (1995)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Stein, E.M.: An example on the Heisenberg group related to the Lewy operator. Invent. Math. 69, 209–216 (1982)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Uguzzoni, F.: Estimates of the Green function for \(X\)-elliptic operators. Math. Ann. 361, 169–190 (2015)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BolognaBolognaItaly

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