Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

On the Grushin operator and hyperbolic symmetry

Author(s): William Beckner
Journal: Proc. Amer. Math. Soc. 129 (2001), 1233-1246.
MSC (2000): Primary 58J70, 35A15
Posted: October 10, 2000
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry.

References:

[1]
W. Beckner, Sobolev inequalities, the Poisson semigroup and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. 89 (1992), 4816-4819. MR 93d:26018

[2]
W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math. 138 (1993), 213-242. MR 94m:58232

[3]
W. Beckner, Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, 1995, pp. 36-68. MR 95m:42004

[4]
W. Beckner, Sharp inequalities and geometric manifolds, J. Fourier Anal. Appl. 3 (1997), 825-836. MR 2000c:58059

[5]
W. Beckner, Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11 (1999), 105-137. MR 2000a:46049

[6]
W. Beckner, Riesz potentials and Sobolev embedding on hyperbolic space, (in preparation).

[7]
W. Beckner, Sobolev inequalities, vortex dynamics and Lie groups, (in preparation).

[8]
H.J. Brascamp, E.H. Lieb and J.M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227-237. MR 49:10835

[9]
G.B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373-376. MR 47:3816

[10]
L.E. Fraenkel, On steady vortex rings with swirl and a Sobolev inequality, Progress in partial differential equations, calculus of variations and applications, Longman Scientific and Technical, 1992, pp. 13-26. MR 93k:76020

[11]
P.R. Garabedian, Partial differential equations, John Wiley, 1964. MR 28:5247

[12]
N. Garofalo and D. Vassilev, Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups, (preprint, April 1999).

[13]
V.V. Grushin, On a class of hypoelliptic operators, Math. Sbornik 12 (1970), 458-475.

[14]
S. Helgason, Differential geometry and symmetric spaces, Academic Press, 1962. MR 26:2986

[15]
S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565-601. MR 29:2323

[16]
D. Jerison and J.M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1-13. MR 89b:53063

[17]
F. John, The fundamental solution of linear elliptic differential equations with analytic coefficients, Comm. Pure Appl. Math. 3 (1950), 273-304. MR 13:40h

[18]
S. Lang, $SL_2(R)$, Addison-Wesley, 1975. MR 55:3170

[19]
H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. 66 (1957), 155-158. MR 19:551d

[20]
F. Mugelli and G. Talenti, Sobolev inequalities in 2-dimensional hyperbolic space, International series of numerical mathematics, Vol. 123, Birkhäuser Verlag, 1997, pp. 201-216. MR 98g:46044

[21]
F. Mugelli and G. Talenti, Sobolev inequalities in 2-d hyperbolic space: a borderline case, J. Inequal. and Appl. 2 (1998), 195-228. CMP 99:08

[22]
A. Nagel and E.M. Stein, Lectures on pseudo-differental operators, Princeton University Press, 1979. MR 82f:47059

[23]
P.G. Saffman, Vortex dynamics, Cambridge University Press, 1992. MR 94c:76015

[24]
N. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and geometry on groups, Cambridge University Press, 1992. MR 95f:43008

[25]
N.J. Vilenkin, Special functions and the theory of group representations, American Mathematical Society, 1968. MR 37:5429

[26]
A. Weil, L'integration dans les groupes topologiques et ses applications, Hermann, 1940. MR 3:198b

[27]
E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge University Press, 1927. Reprint MR 97k:01072

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58J70, 35A15

Retrieve articles in all Journals with MSC (2000): 58J70, 35A15

Additional Information:

William Beckner
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082
Email: beckner@math.utexas.edu

DOI: 10.1090/S0002-9939-00-05630-6
PII: S 0002-9939(00)05630-6
Received by editor(s): March 19, 1999
Received by editor(s) in revised form: July 2, 1999
Posted: October 10, 2000
Additional Notes: This work was partially supported by the National Science Foundation.
Communicated by: Christopher D. Sogge
Copyright of article: Copyright 2000, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google