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An Extension of Hörmander’s Hypoellipticity Theorem

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Abstract

Motivated by applications to stochastic differential equations, an extension of Hörmander’s hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-wise Bessel kernel estimates and a weighted Sobolev inequality of Stein and Weiss. Of particular interest is that our results apply to operators with quite general first-order terms.

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References

  1. Aronszajn, N., Smith, K.T.: Theory of Bessel potentials. I. Ann. Inst. Fourier (Grenoble) 11, 385–475 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bismut, J.-M.: Martingales, the Malliavin calculus and Hörmander’s theorem. In: Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980), Volume 851 of Lecture Notes in Math., pp. 85–109. Springer, Berlin (1981)

    Google Scholar 

  3. Bismut, J.-M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Z. Wahrsch. Verw. Gebiete 56(4), 469–505 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. 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), vi+123 (2010)

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  6. Cass, T.: Smooth densities for solutions to stochastic differential equations with jumps. Stoch. Process. Appl. 119(5), 1416–1435 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Coifman, R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28(3), xi, 177–202 (1978)

    Article  MathSciNet  Google Scholar 

  8. Fefferman, C., Phong, D. H.: Subelliptic eigenvalue problems. In: Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pp 590–606. Wadsworth, Belmont (1983)

    Google Scholar 

  9. Franchi, B.: Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. Trans. Am. Math. Soc. 327(1), 125–158 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10(4), 523–541 (1983)

    MATH  MathSciNet  Google Scholar 

  11. Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. (2) 164(3), 993–1032 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Hörmander, L.: The analysis of linear partial differential operators. III. Classics in Mathematics. Springer, Berlin, 2007. Pseudo-differential operators, Reprint of the 1994 edition

  14. Ikeda, N., Watanabe, S.: An introduction to Malliavin’s calculus. In: Stochastic Analysis (Katata/Kyoto, 1982), Volume 32 of North-Holland Math. Library, pp 1–52. North-Holland, Amsterdam (1984)

    Google Scholar 

  15. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. II. J. Facil. Sci. Univ. Tokyo Sect. IA Math. 32(1), 1–76 (1985)

    MATH  MathSciNet  Google Scholar 

  17. Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. III. J. Facil. Sci. Univ. Tokyo Sect. IA Math. 34(2), 391–442 (1987)

    MATH  MathSciNet  Google Scholar 

  18. Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus, I. In: Stochastic Analysis (Katata/Kyoto, 1982), Volume 32 of North-Holland Math. Library, pp 271–306. North-Holland, Amsterdam (1984)

    Google Scholar 

  19. Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. In: Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), pp. 195–263. Wiley, New York (1978)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Montanari, A., Morbidelli, D.: Nonsmooth Hörmander vector fields and their control balls. Trans. Am. Math. Soc. 364(5), 2339–2375 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Norris, J.: Simplified Malliavin calculus. In: Séminaire de Probabilités, XX, 1984/85, Volume 1204 of Lecture Notes in Math., pp 101–130. Springer, Berlin (1986)

    Google Scholar 

  23. Ocone, D.: Stochastic calculus of variations for stochastic partial differential equations. J. Funct. Anal. 79(2), 288–331 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sanz-Solé, M.: Malliavin Calculus. Fundamental Sciences. EPFL Press, Lausanne (2005). With applications to stochastic partial differential equations

    Google Scholar 

  25. Sawyer, E.T., Wheeden, R.L.: Hölder continuity of weak solutions to subelliptic equations with rough coefficients. Mem. Am. Math. Soc. 180(847), x+157 (2006)

    MathSciNet  Google Scholar 

  26. Stein, E.M., Weiss, G.: Fractional integrals on n-dimensional Euclidean space. J. Math. Mech. 7, 503–514 (1958)

    MATH  MathSciNet  Google Scholar 

  27. Elias, M.: Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)

    Google Scholar 

  28. Stroock, D.W.: Partial differential equations for probabilists, Volume 112 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012). Paperback edition of the 2008 original

    Google Scholar 

  29. Wang, L.: Hölder estimates for subelliptic operators. J. Funct. Anal. 199(1), 228–242 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Drake University, Des Moines, IA, 50311, USA

    David P. Herzog

  2. Department of Mathematics and Statistics, University of Massachusetts, Lederle Graduate Research Tower, Amherst, MA, 01003, USA

    Nathan Totz

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  1. David P. Herzog
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  2. Nathan Totz
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Correspondence to David P. Herzog.

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Herzog, D.P., Totz, N. An Extension of Hörmander’s Hypoellipticity Theorem. Potential Anal 42, 403–433 (2015). https://doi.org/10.1007/s11118-014-9439-0

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  • DOI: https://doi.org/10.1007/s11118-014-9439-0

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