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Estimates for powers of sub-Laplacian on the non-isotropic Heisenberg group

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Abstract

Assume that \({\mathcal{L}}_\alpha = - \frac{1}{2}\sum\nolimits_{j = 1}^n {(Z_j \bar Z_j + \bar Z_j Z_j ) + i\alpha T} \) is the sub-Laplacian on the nonisotropic Heisenberg group H n ;Z j ,Z j for j = 1, 2, …,n and T are the basis of the Lie algebra h n .We apply the Laguerre calculus to obtain the explicit kernel for the fundamental solution of the powers of L α and the heat kernel exp{−sL α }.Estimates for this kernel in various function spaces can be deduced easily.

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References

  1. Beals, R., Gaveau, B., and Greiner, P.C. Complex Hamillonian mechanics and parametrices for subelliptic Laplacians,Bull. Sci. Math.,121, 1–36, 97–149, 195–259, (1997).

    MathSciNet  MATH  Google Scholar 

  2. Beals, R., Gaveau, B., Greiner, P.C., and Vauthier, J. The Laguerre calculus on the Heisenberg group: II,Bull. Sci. Math.,110, 225–288, (1986).

    MathSciNet  MATH  Google Scholar 

  3. Beals, R. and Greiner, P.C.Calculus on Heisenberg Manifolds, vol. 119 of Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1988.

    Google Scholar 

  4. Benson, C., Dolley, A.H., and Ratcliff, G. Fundamental solutions for the powers of the Heisenberg sub-Laplacian,Illinois J. Math.,37, 455–476, (1993).

    MathSciNet  MATH  Google Scholar 

  5. Chang, D.-C. Estimates for the singular integral operators with mixed type homogeneities in Hardy classes,Math. Ann.,287, 303–322, (1990).

    Article  MathSciNet  MATH  Google Scholar 

  6. Chang, D.-C., Dafni, G., and Stein, E.M. Hardy spaces,BMO, and boundary value problems for the Laplacian on a smooth domain inR N, to appear inTrans. AMS, (1998).

  7. Chang, D.-C., Krantz, S.G., and Stein, E.M.H p theory on a smooth domain inR N and elliptic boundary value problems,J. Funct. Anal.,114, 286–347, (1993).

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang, D.-C. and Tie, J. Estimates for the spectrum projection operators of the sub-Laplacian on the Heisenberg group,J. d’Analyse Math.,71, 315–347, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  9. Chang, D.-C. and Tie, J. Some differential operators related to the Heisenberg sub-Laplacian, to appearMath. Nachr., (1999).

  10. Christ, M. and Geller, D. Singular integral characterizations of Hardy spaces on homogeneous groups,Duke Math. J.,51, 547–598, (1984).

    Article  MathSciNet  MATH  Google Scholar 

  11. Folland, G.B.Harmonic Analysis in Phase Space. vol. 122 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1989.

    MATH  Google Scholar 

  12. Folland, G.B. and Stein, E.M.Hardy Spaces on Homogeneous Groups, vol. 28 of Mathematical Notes Series, Princeton University Press, Princeton, NJ, 1982.

    MATH  Google Scholar 

  13. Folland, G.B. and Stein, E.M. Estimates for the\(\bar \partial _b - complex\) and analysis on the Heisenberg group,Comm. Pure Appl. Math.,27, 429–522, (1974).

    Article  MathSciNet  MATH  Google Scholar 

  14. Gaveau, B. Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents,Acta Math.,139, 95–153, (1977).

    Article  MathSciNet  MATH  Google Scholar 

  15. Gel’fand, I.M. and Shilov, G.E.Generalized Functions, vol. 1, Academic Press, New York, 1964.

    MATH  Google Scholar 

  16. Geller, D. Fourier analysis on the Heisenberg group,Proc. Nat. Acad. Sci. USA,74, 1328–1331, (1977).

    Article  MathSciNet  MATH  Google Scholar 

  17. Goldberg, D. A local version of real Hardy spaces,Duke Math. J.,46, 27–42, (1979).

    Article  MathSciNet  MATH  Google Scholar 

  18. Greiner, P.C., On the Laguerre calculus of left-invariant convolution (pseudo-differential) operators on Heisenberg group,Séminaire Goulaouic-Meyer-Schwartz,XI, 1–39, (1980–1981).

    Google Scholar 

  19. Hulanicki, A. The distribution of energy in the Brownian motion in the Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group,Studia Math.,56, 165–173, (1976).

    MathSciNet  MATH  Google Scholar 

  20. Krantz, S.G. Analysis on the Heisenberg group and estimates for functions in Hardy classes of several complex variables.Math. Ann.,244, 243–262, (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Müller, D. and Stein, E.M. On spectral multipliers for Heisenberg and related groups,J. Math. Pures Appl.,73, 413–440, (1994).

    MathSciNet  MATH  Google Scholar 

  22. Peetre, J. The Weyl transform and Laguerre polynomials,Le Matematiche,27, 301–323, (1972).

    MathSciNet  Google Scholar 

  23. Stein, E.M.Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Georgetown University, 20057, Washington, D.C.

    Der-Chen Chang & Jingzhi Tie

  2. Department of Mathematics, University of Georgia, 30602, Athene, GA

    Der-Chen Chang & Jingzhi Tie

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  1. Der-Chen Chang
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  2. Jingzhi Tie
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Chang, DC., Tie, J. Estimates for powers of sub-Laplacian on the non-isotropic Heisenberg group. J Geom Anal 10, 653–678 (2000). https://doi.org/10.1007/BF02921990

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  • DOI: https://doi.org/10.1007/BF02921990

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