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Heat Kernels in Sub-Riemannian Settings

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Geometric Analysis and PDEs

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1977))

Abstract

In this lectures we present a series of results concerning a class of diffusion second order PDE’s of heat-type. The results we show have been obtained in collaboration with M.Bramanti, L.Brandolini and F. Uguzzoni (see [9], [10], [24]). The exended version of the main results presented in these notes is contained in [10].

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  1. S. Kusuoka, D. Stroock: Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 1–76.

    MathSciNet  MATH  Google Scholar 

  2. S. Kusuoka, D. Stroock: Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391–442.

    MathSciNet  MATH  Google Scholar 

  3. S. Kusuoka, D. Stroock: Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator. Ann. of Math. (2) 127 (1988), no. 1, 165–189.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Lanconelli, S. Polidoro: On a class of hypoelliptic evolution operators. Partial differential equations, II (Turin, 1993). Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 1, 29–63.

    MathSciNet  MATH  Google Scholar 

  5. E. Lanconelli, F. Uguzzoni: Potential Analysis for a Class of Diffusion Equations: a Gaussian Bounds Approach, Preprint.

    Google Scholar 

  6. E. E. Levi: Sulle equazioni lineari totalmente ellittiche alle derivate parziali. Rend. Circ. Mat. Palermo 24, 275–317 (1907) (Also with corrections in: Eugenio Elia Levi, Opere, vol. II, 28–84. Roma, Edizioni Cremonese 1960).

    Article  MATH  Google Scholar 

  7. E. E. Levi: I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali, Memorie Mat. Fis. Soc. Ital. Scienza (detta dei XL) (3) 16, 3–113 (1909). (Also with corrections in: Eugenio Elia Levi, Opere, vol. II, 207–343. Roma, Edizioni Cremonese 1960).

    Google Scholar 

  8. V. Martino, A. Montanari : Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems, Preprint.

    Google Scholar 

  9. A. Montanari: Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation. Comm. Partial Differential Equations 26 (2001), no. 9–10, 1633–1664.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Montanari, E. Lanconelli: Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems. J. Differential Equations 202 (2004), no. 2, 306–331.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Nagel, E. M. Stein, S. Wainger: Balls and metrics defined by vector fields I: Basic properties. Acta Mathematica, 155 (1985), 130–147.

    Article  MathSciNet  Google Scholar 

  12. J. Nash: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958) 931–954.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Picone: Maggiorazione degli integrali di equazioni lineari ellittico-paraboliche alle derivate parziali del secondo ordine. Rend. Accad. Naz. Lincei 5, n.6 (1927) 138–143.

    MATH  Google Scholar 

  14. S. Polidoro: On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Le Matematiche (Catania) 49 (1994), no. 1, 53–105.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. L. Saloff-Coste, Aspects of Sobolev-type inequalities. London Mathematical Society Lecture Note Series, 289. Cambridge University Press, Cambridge, 2002.

    Google Scholar 

  17. L. Saloff-Coste, D. W. Stroock: Opérateurs uniformément sous-elliptiques sur les groupes de Lie. J. Funct. Anal. 98 (1991), no. 1, 97–121.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Sanchez-Calle: Fundamental solutions and geometry of sum of squares of vector fields. Inv. Math., 78 (1984), 143–160.

    Article  MathSciNet  MATH  Google Scholar 

  19. Z. Slodkowski, G. Tomassini: Weak solutions for the Levi equation and envelope of holomorphy. J. Funct. Anal. 101 (1991), no. 2, 392–407.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. N. Th. Varopoulos: Théorie du potentiel sur les groupes nilpotents. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 5, 143–144.

    MathSciNet  MATH  Google Scholar 

  22. N. Th. Varopoulos, Analysis on nilpotent groups. J. Funct. Anal. 66 (1986), no. 3, 406–431.

    Article  MathSciNet  MATH  Google Scholar 

  23. N. Th. Varopoulos, L. Saloff-Coste, T. Coulhon: Analysis and geometry on groups. Cambridge Tracts in Mathematics, 100. Cambridge University Press, Cambridge, 1992.

    Google Scholar 

  24. C. J. Xu: Regularity for quasilinear second-order subelliptic equations. Comm. Pure Appl. Math. 45 (1992), no. 1, 77–96.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Universita’ degli Studi di Bologna, P.zza di Porta S. Donato, 5 40127, Bologna, Italy

    Ermanno Lanconelli

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  1. Ermanno Lanconelli
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Correspondence to Ermanno Lanconelli .

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Editors and Affiliations

  1. Dept. Mathematics, Princeton University, Washington Rd., Princeton, 08544-1000, U.S.A.

    Sun-Yung Alice Chang

  2. Department of Mathematics, SISSA, Via Beirut 2-4, Trieste, 34014, Italy

    Antonio Ambrosetti

  3. Studi Avanzati (SISSA), Scuola Internazionale Superiore di, via Beirut 2-4, Trieste, 34014, Italy

    Andrea Malchiodi

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Lanconelli, E. (2009). Heat Kernels in Sub-Riemannian Settings. In: Chang, SY., Ambrosetti, A., Malchiodi, A. (eds) Geometric Analysis and PDEs. Lecture Notes in Mathematics(), vol 1977. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01674-5_2

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