On the Grushin operator and hyperbolic symmetry

Author:
William Beckner

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1233-1246

MSC (2000):
Primary 58J70, 35A15

DOI:
https://doi.org/10.1090/S0002-9939-00-05630-6

Published electronically:
October 10, 2000

MathSciNet review:
1709740

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**W. Beckner,*Sobolev inequalities, the Poisson semigroup and analysis on the sphere*, 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, , 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**

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Additional Information

**William Beckner**

Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082

Email:
beckner@math.utexas.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05630-6

Received by editor(s):
March 19, 1999

Received by editor(s) in revised form:
July 2, 1999

Published electronically:
October 10, 2000

Additional Notes:
This work was partially supported by the National Science Foundation.

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 2000
American Mathematical Society