Revista Matemática Iberoamericana

High order regularity for subelliptic operators on Lie groups of polynomial growth

Nick Dungey

Full-text: Open access

Abstract

Let $G$ be a Lie group of polynomial volume growth, with Lie algebra $\mbox{\gothic g}$. Consider a second-order, right-invariant, subelliptic differential operator $H$ on $G$, and the associated semigroup $S_t = e^{-tH}$. We identify an ideal $\mbox{\gothic n}'$ of $\mbox{\gothic g}$ such that $H$ satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of $\mbox{\gothic n}'$. The regularity is expressed as $L_2$ estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in $L_p$, $1<p<\infty$, of some associated Riesz transform operators. Finally, we show that $\mbox{\gothic n}'$ is the largest ideal of $\mbox{\gothic g}$ for which the regularity results hold. Various algebraic characterizations of $\mbox{\gothic n}'$ are given. In particular, $\mbox{\gothic n}'= \mbox{\gothic s}\oplus \mbox{\gothic n}$ where $\mbox{\gothic n}$ is the nilradical of $\mbox{\gothic g}$ and $\mbox{\gothic s}$ is the largest semisimple ideal of $\mbox{\gothic g}$. Additional features of this article include an exposition of the structure theory for $G$ in Section 2, and a concept of twisted multiplications on Lie groups which includes semidirect products in the Appendix.

Article information

Source
Rev. Mat. Iberoamericana, Volume 21, Number 3 (2005), 929-996.

Dates
First available in Project Euclid: 11 January 2006

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1136999137

Mathematical Reviews number (MathSciNet)
MR2232672

Zentralblatt MATH identifier
1099.22007

Subjects
Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]
Secondary: 35B65: Smoothness and regularity of solutions 58J35: Heat and other parabolic equation methods

Keywords
Lie group subelliptic operator heat kernel Riesz transform regularity estimates

Citation

Dungey, Nick. High order regularity for subelliptic operators on Lie groups of polynomial growth. Rev. Mat. Iberoamericana 21 (2005), no. 3, 929--996. https://projecteuclid.org/euclid.rmi/1136999137


Export citation

References

  • Alexopoulos, G.: An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Can. J. Math. 44 (1992), 691-727.
  • Alexopoulos, G.: Sub-Laplacians with drift on Lie groups of polynomial volume growth. Mem. Amer. Math. Soc. 155 (2002), no. 739.
  • Auscher, P., Elst, A.F.M. ter and Robinson, D.W.: On positive Rockland operators. Colloq. Math. 67 (1994), 197-216.
  • Barnes, D.W.: On Cartan subalgebras of Lie algebras. Math. Z., 101: 350-355, 1967.
  • Bourbaki, N.: Elements de mathematique: Groupes and algebres de Lie. Hermann, Paris, 1975.
  • Brocker, T. and Dieck, T. tom: Representations of compact Lie groups. Graduate Texts in Mathematics 98. Springer-Verlag, New York, 1985.
  • Butzer, P.L. and Berens, H.: Semi-groups of operators and approximation. Die Grundlehren der mathematischen Wissenschaften 145. Springer-Verlag, Berlin, 1967.
  • Coifman, R.R. and Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogénes. Lect. Notes in Math 242. Springer-Verlag, Berlin, 1971.
  • Coifman, R.R. and Weiss, G.: Transference methods in analysis. CBMS Regional Conference Series in Mathematics 31. Amer. Math. Soc., Providence, 1977.
  • Corduneanu, C.: Almost-periodic functions. Interscience Publishers, New York, 1968.
  • Corwin, L. and Greenleaf, F.P.: Representations of nilpotent Lie groups and their applications: Part I. Cambridge Studies in Advanced Mathematics 18. Cambridge University Press, 1990.
  • David, G. and Journé, J.L.: A boundedness criterion for generalised Calderón-Zygmund operators. Ann. of Math.(2) 120 (1984), 371-397.
  • Dungey, N.: Higher order operators and Gaussian bounds on Lie groups of polynomial growth. J. Operator Theory 46 (2001), 45-61.
  • Dungey, N., Elst, A.F.M. ter and Robinson, D.W.: Asymptotics of sums of subcoercive operators. Colloq. Math. 82 (1999), 231-260.
  • Dungey, N., Elst, A.F.M. ter and Robinson, D.W.: Analysis on Lie groups with polynomial growth. Birkhauser, Boston, 2003.
  • Dziubanski, J., Hebisch, W. and Zienkiewicz, J.: Note on semigroups generated by positive Rockland operators on graded homogeneous groups. Studia Math. 110 (1994), 115-126.
  • Elst, A.F.M. ter and Robinson, D.W.: Subelliptic operators on Lie groups: regularity. J. Austral. Math. Soc. Ser.A 57 (1994), 179-229.
  • Elst, A.F.M. ter and Robinson, D.W.: Subcoercivity and subelliptic operators on Lie groups I: Free nilpotent groups. Potential Anal. 3 (1994), 283-337.
  • Elst, A.F.M. ter and Robinson, D.W.: Local lower bounds on heat kernels. Positivity 2 (1998), 123-151.
  • Elst, A.F.M. ter and Robinson, D.W.: Weighted subcoercive operators on Lie groups. J. Funct. Anal. 157 (1998), 88-163.
  • Elst, A.F.M. ter and Robinson, D.W.: Gaussian bounds for complex subelliptic operators on Lie groups of polynomial growth. Bull. Austral. Math. Soc. 67 (2003), 201-218.
  • Elst, A.F.M. ter; Robinson, D.W. and Sikora, A.: Heat kernels and Riesz transforms on nilpotent Lie groups. Colloq. Math. 74 (1997), 191-218.
  • Elst, A.F.M. ter; Robinson, D.W. and Sikora, A.: Riesz transforms and Lie groups of polynomial growth. J. Funct. Anal. 162 (1999), 14-51.
  • Folland, G.B.: A course in abstract harmonic analysis. CRC Press, Florida, 1995.
  • Folland, G.B. and Stein, E.S.: Hardy spaces on homogeneous groups. Mathematical Notes 28. Princeton University Press, Princeton, 1982.
  • Guivarc'h, Y.: Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 333-379.
  • Helffer, B. and Nourrigat, J.: Caractérisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué. Comm. Partial Differential Equations 4 (1979), 899-958.
  • Kato T.: Perturbation theory for linear operators, second edition. Die Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag, Berlin, 1984.
  • Nagel, A., Ricci, F. and Stein, E.M.: Harmonic analysis and fundamental solutions on nilpotent Lie groups. In Analysis and partial differential equations, 249-275. Lecture Notes in Pure and Appl. Math. 122. Marcel Dekker, New York, 1990.
  • Robinson, D.W.: Elliptic operators and Lie groups. Oxford University Press, Oxford, 1991.
  • Saloff-Coste, L.: Analyse sur les groupes de Lie à croissance polynômiale. Ark. Mat. 28 (1990), 315-331.
  • Varadarajan, V.S.: Lie Groups, Lie Algebras, and their Representations. Graduate Texts in Mathematics 102. Springer-Verlag, New York, 1984.
  • Varopoulos, N.T.: Analysis on nilpotent groups. J. Funct. Anal. 66 (1986), 406-431.
  • Varopoulos, N.T.: Analysis on Lie groups. J. Funct. Anal. 76 (1988), 346-410.
  • Varopoulos, N.T., Saloff-Coste, L. and Coulhon, T.: Analysis and geometry on groups. Cambridge Tracts in Mathematics 100. Cambridge University Press, Cambridge, 1992.
  • Winter, D.J.: Abstract Lie algebras. The M.I.T. Press, Cambridge, Mass.-London, 1972.