Skip to main content
SpringerLink
Log in

Beyond Hörmander’s Operators

  • Chapter
  • First Online:
An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 1255 Accesses

Abstract

In this last chapter I want to discuss some developments which have taken place in the study of Hörmander’s operators and related topics since the 1990’s. As we will see, most of these developments have extended the class of operators under study, passing from classical Hörmander’s operators to operators “structured on Hörmander’s vector fields”, in various senses, or operators also containing nonsmooth ingredients. In the end, most of these new kinds of operators are no longer hypoelliptic, but still share with classical Hörmander’s operators several deep properties.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The first of the two papers actually deals with a slightly simplified equation, but already contains the main ideas exploited in the second one to handle the Levi equation.

  2. 2.

    What follows in this paragraph is extracted from the introduction of [13].

  3. 3.

    The following paragraph is extracted from the Introduction of [15].

References

  1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary of solutions of elliptic partial differential equations under general boundary conditions. Part I, Comm. Pure Applied Math. 12, 623–727 (1959); Part II, Comm. Pure Applied Math. 17(1), 35–92 (1964)

    Google Scholar 

  2. Bedford, E., Gaveau, B.: Hypersurfaces with bounded Levi form. Indiana Univ. Math. J. 27(5), 867–873 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups. Adv. Differ. Equ. 7, 1153–1192 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Fundamental solutions for non-divergence form operators on stratified groups. Trans. Am. Math. Soc. 356(7), 2709–2737 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonfiglioli, A., Uguzzoni, F.: Families of diffeomorphic sub-Laplacians and free Carnot groups. Forum Math. 16(3), 403–415 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonfiglioli, A., Uguzzoni, F.: Harnack inequality for non-divergence form operators on stratified groups. Trans. Am. Math. Soc. 359, 2463–2481 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bramanti, M.: Singular integrals in nonhomogeneous spaces: \( L^{2}\) and \( L^{p}\) continuity from Hölder estimates. Revista Matematica Iberoamericana 26(1), 347–366 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bramanti, L., Brandolini, M.: \( L^{ p}\) -estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Rend. Sem. Mat. dell’Univ. e del Politec. di Torino. 58(4), 389–433 (2000)

    Google Scholar 

  9. Bramanti, M., Brandolini, L.: \( L^{ p}\) -estimates for nonvariational hypoelliptic operators with VMO coefficients. Trans. Am. Math. Soc. 352(2), 781–822 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bramanti, M., Brandolini, L.: Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs. Rev. Mat. Iberoamericana 21(2), 511–556 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bramanti, M., Brandolini, L.: Schauder estimates for parabolic nondivergence operators of Hörmander type. J. Differ. Equ. 234(1), 177–245 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bramanti, M., Brandolini, L., Lanconelli, E., Uguzzoni, F.: Heat kernels for non-divergence operators of Hörmander type. Comptes rendus - Mathematique. 343(7), 463–466 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bramanti, M., Brandolini, L., Lanconelli, E., Uguzzoni, F.: Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities. Mem. AMS 204(961), 1–136 (2010)

    MathSciNet  Google Scholar 

  14. Bramanti, M., Brandolini, L., Manfredini, M., Pedroni, M.: Fundamental solutions and local solvability for nonsmooth Hörmander’s operators. Submitted, Preprint (2013). http://arxiv.org/abs/1305.3398

  15. Bramanti, M., Brandolini, L., Pedroni, M.: Basic properties of nonsmooth Hörmander’s vector fields and Poincaré’s inequality. Forum Mathematicum. 25(4), 703–769 (2013)

    Google Scholar 

  16. Bramanti, M., Brandolini, L., Pedroni, M.: On the lifting and approximation theorem for nonsmooth vector fields. Indiana Univ. Math. J. 59(6), 1889–1934 (2010)

    Google Scholar 

  17. Bramanti, M., Cerutti, M.C.: Commutators of singular integrals on homogeneous spaces. Boll. Un. Mat. Ital. B (7) 10(4), 843–883 (1996)

    Google Scholar 

  18. Bramanti, M., Cerutti, M.C.: \( W_{p}^{1,2}\) -solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. Comm. Partial Differ. Equ. 18(9–10), 1735–1763 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bramanti, M., Cerutti, M.C., Manfredini, M.: \(L^{p}\) -estimates for some ultraparabolic operators with discontinuous coefficients. J. Math. Anal. Appl. 200(2), 332–354 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bramanti, M., Cupini, G., Lanconelli, E., Priola, E.: Global \( L^{ p}\) estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients. Mathematische Nachrichten, 286(11–12), 1087–1101 (2013)

    Google Scholar 

  21. Bramanti, M., Cupini, G., Lanconelli, E., Priola, E.: Global \( L^{ p}\) estimates for degenerate Ornstein-Uhlenbeck operators. Mathematische Zeitschrift 266(4), 789–816 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bramanti, M., Fanciullo, M.S.: BMO estimates for nonvariational operators with discontinuous coefficients structured on Hörmander’s vector fields on Carnot groups. Advances in Differential Equations 18(9–10), 955–1004 (2013)

    Google Scholar 

  23. Bramanti, M., Zhu, M.: \( L^{ p}\) and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift. Anal. Partial Differ. Equ. ArXiv: 1103.5116v1 26. (2011, to appear)

    Google Scholar 

  24. Bramanti, M., Zhu, M.: Local real analysis in locally homogeneous spaces. Manuscripta Math. 138(3–4), 477–528 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155(3–4), 261–301 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Capogna, L., Han, Q.: Pointwise Schauder estimates for second order linear equations in Carnot groups. Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), pp. 45–69, Contemp. Math., 320, Amer. Math. Soc., Providence, RI (2003)

    Google Scholar 

  27. Chiarenza, F., Frasca, M., Longo, P.: Interior \( W^{2, p}\)-estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat. 40, 149–168 (1991)

    MathSciNet  MATH  Google Scholar 

  28. Chiarenza, F., Frasca, M., Longo, P.: \( W^{2, p}\) -solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336(1), 841–853 (1993)

    MathSciNet  Google Scholar 

  29. Citti, G.: \(C^{\infty } \) regularity of solutions of a quasilinear equation related to the Levi operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(3), 483–529 (1996)

    Google Scholar 

  30. Citti, G.: \(C^{\infty }\) regularity of solutions of the Levi equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(4), 517–534 (1998)

    Google Scholar 

  31. Citti, G., Lanconelli, E., Montanari, A.: Smoothness of Lipchitz-continuous graphs with nonvanishing Levi curvature. Acta Math. 188(1), 87–128 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Citti, G., Montanari, A.: \(C^{\infty } \) regularity of solutions of an equation of Levi’s type in \({\mathbb{R}}^{2n+1}\). Ann. Mat. Pura Appl. (4) 180(1), 27–58 (2001)

    Google Scholar 

  33. Coifman,R.R., Rochberg, G.: Weiss: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)

    Google Scholar 

  34. Di Francesco, M., Polidoro, S.: Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Differ. Equ. 11(11), 1261–1320 (2006)

    MATH  Google Scholar 

  35. Fabes, E.B., Stroock, D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96, 327–338 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Fefferman, C., Sánchez-Calle, A.: Fundamental solutions for second order subelliptic operators. Ann. Math. (2) 124(2), 247–272 (1986)

    Google Scholar 

  37. Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(2), 161–207 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  38. Franchi, B.: Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. Trans. Am. Math. Soc. 327, 125–158 (1991)

    MathSciNet  MATH  Google Scholar 

  39. Gutiérrez, C.E., Lanconelli, E.: Schauder estimates for sub-elliptic equations. J. Evol. Equ. 9(4), 707–726 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  41. Huisken, G., Klingenberg, W.: Flow of real hypersurfaces by the trace of the Levi form. Math. Res. Lett. 6(5–6), 645–661 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jerison, D.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53(2), 503–523 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. Jerison, D., Sánchez-Calle, A.: Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. Math. J. 35(4), 835–854 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  44. Karmanova, M., Vodopyanov, S.: Geometry of Carnot-Carathéodory spaces, differentiability, coarea and area formulas. Anal. Math. Phys. Trends Math., pp. 233–335 (2009)

    Google Scholar 

  45. Kohn, J.J.: Pseudo-differential operators and hypoellipticity. Partial differential equations (Proc. Sympos. Pure Math., vol. XXIII, Univ. California, Berkeley, Calif., 1971), pp. 61–69. American Mathematical Society, Providence (1973)

    Google Scholar 

  46. Lanconelli, E.: Heat Kernels in Subriemannian settings. In: Proceeding of the CIME Summer Course 2007 on “Geometric Analysis and PDEs”. Springer Lectures Notes in Math. vol. 2009, pp. 35–61 (1977)

    Google Scholar 

  47. Lanconelli, E., Lascialfari, F.: A boundary value problem for a class of quasilinear operators of Fokker-Planck type. In: Proceedings of the Conference “Differential Equations”. Ann. Univ. Ferrara Sez. VII (N.S.) 41 (1996), suppl., 65–84 (1997)

    Google Scholar 

  48. Lanconelli, E., Morbidelli, D.: On the Poincaré inequality for vector fields. Ark. Mat. 38(2), 327–342 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lanconelli, E., Pascucci, A., Polidoro, S.: Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. Nonlinear problems in mathematical physics and related topics, II, 243–265, Int. Math. Ser. (NY), 2. Kluwer/Plenum, New York (2002)

    Google Scholar 

  50. Lanconelli, E., Polidoro, S.: On a class of hypoelliptic evolution operators. Partial differential equations, II (Turin, 1993). Rend. Sem. Mat. Univ. Politec. Torino 52(1), 29–63 (1994)

    MathSciNet  MATH  Google Scholar 

  51. Manfredini, M.: The Dirichlet problem for a class of ultraparabolic equations. Adv. Differ. Equ. 2(5), 831–866 (1997)

    MathSciNet  MATH  Google Scholar 

  52. Manfredini, M., Polidoro, S.: Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1(3), 651–675 (1998)

    Google Scholar 

  53. Maugeri, A., Palagachev, D.K., Softova, L.G.: Elliptic and parabolic equations with discontinuous coefficients. Math. Res. vol. 109, Wiley (2000)

    Google Scholar 

  54. Miranda, C.: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui. Ann. Mat. Pura Appl. (4) 63, 353–386 (1963)

    Google Scholar 

  55. Montanari, A.: Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation. Comm. Partial Differ. Equ. 26(9–10), 1633–1664 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  56. Montanari, A., Lanconelli, E.: Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems. J. Differ. Equ. 202(2), 306–331 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  57. Montanari, A., Lascialfari, F.: The Levi Monge-Ampère equation: smooth regularity of strictly Levi convex solutions. J. Geom. Anal. 14(2), 331–353 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  58. Montanari, A., Morbidelli, D.: A Frobenius-type theorem for singular Lipschitz distributions. J. Math. Anal. Appl. 399(2), 692–700 (2013)

    Google Scholar 

  59. Montanari, A., Morbidelli, D.: Sobolev and Morrey estimates for non-smooth vector fields of step two. Z. Anal. Anwendungen 21(1), 135–157 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  60. Montanari, A., Morbidelli, D.: Balls defined by nonsmooth vector fields and the Poincaré inequality. Ann. Inst. Fourier (Grenoble) 54(2), 431–452 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  61. Montanari, A., Morbidelli, D.: Nonsmooth Hörmander vector fields and their control balls. Trans. Am. Math. Soc 364, 2339–2375 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  62. Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155(1–2), 103–147 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  63. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  64. Pascucci, A., Polidoro, S.: A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations. J. Math. Anal. Appl. 282(1), 396–409 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  65. Pascucci, A., Polidoro, S.: The Moser’s iterative method for a class of ultraparabolic equations. Commun. Contemp. Math. 6(3), 395–417 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  66. Pascucci, A., Polidoro, S.: On the Harnack inequality for a class of hypoelliptic evolution equations. Trans. Am. Math. Soc. 356(11), 4383–4394 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  67. Polidoro, S.: On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Le Matematiche (Catania) 49(1), 53–105 (1995)

    Google Scholar 

  68. Polidoro, S.: Uniqueness and representation theorems for solutions of Kolmogorov-Fokker-Planck equations. Rend. Mat. Appl. (7), 15 (1995), no. 4, 535–560 (1996)

    Google Scholar 

  69. Polidoro, S.: A global lower bound for the fundamental solution of Kolmogorov-Fokker-Planck equations. Arch. Ration. Mech. Anal. 137(4), 321–340 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  70. Polidoro, S., Ragusa, M.A.: Sobolev-Morrey spaces related to an ultraparabolic equation. Manuscripta Math. 96(3), 371–392 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  71. Polidoro, S., Ragusa, M.A.: Hölder regularity for solutions of ultraparabolic equations in divergence form. Potential Anal. 14(4), 341–350 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  72. Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3–4), 247–320 (1976)

    Article  MathSciNet  Google Scholar 

  73. Sánchez-Calle, A.: Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78(1), 143–160 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  74. Sarason, D.: Functions of vanishing mean oscillations. Trans. Am. Math. Soc. 207, 391–405 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  75. Sawyer, E.T., Wheeden, R.L.: Hölder continuity of weak solutions to subelliptic equations with rough coefficients. Mem. Am. Math. Soc. 180, 847 (2006)

    Google Scholar 

  76. Slodkowski, Z., Tomassini, G.: Weak solutions for the Levi equation and envelope of holomorphy. J. Funct. Anal. 101(2), 392–407 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  77. Tomassini, G.: Geometric properties of solutions of the Levi-equation. Ann. Mat. Pura Appl. (4) 152, 331–344 (1988)

    Google Scholar 

  78. Xu, C.J.: Regularity for quasilinear second-order subelliptic equations. Comm. Pure Appl. Math. 45(1), 77–96 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

    Marco Bramanti

Authors
  1. Marco Bramanti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Bramanti .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 The Author(s)

About this chapter

Cite this chapter

Bramanti, M. (2014). Beyond Hörmander’s Operators. In: An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-02087-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation