×

zbMATH — the first resource for mathematics

Analysis on geodesic balls of sub-elliptic operators. (Analyse sur les boules d’un opérateur sous-elliptique.) (French) Zbl 0836.35106
We consider a geodesic ball \(B\) associated with a sub-elliptic second order differential operator and obtain sharp Sobolev inequalities and eigenvalue estimates for the Neumann problem on \(B\). Our approach bases on a previous work of D. Jerison. We recover and extend some recent results of G. Lu. The proofs are based on a Whitney covering argument where a family of coverings at different scales is used. This technique allows us to deduce a Sobolev inequality on \(B\) from Poincaré inequalities on smaller balls. By the same token, we give a two-sided estimate of the Weil counting function of the Neumann eigenvalues in \(B\). The bound we obtain for the counting function is more precise than the one resulting directly from the Sobolev inequality. These results apply to the Laplace-Beltrami operators of Riemannian manifolds with Ricci curvature bounded below and to invariant sub-elliptic operators on Lie groups.

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35J70 Degenerate elliptic equations
58J05 Elliptic equations on manifolds, general theory
26D10 Inequalities involving derivatives and differential and integral operators
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bakry, Ledoux, Coulhon et Saloff-Coste.: Sobolev inequalities in disguise, 1994
[2] Biroli M., Mosco U.: Forme de Dirichlet et estimations structurelles dans les milieux discontinus. C.R. Acad. Sci. Paris,313 (1991), 593-598 · Zbl 0760.49004
[3] Buser P.: A note on the isoperimetric constant. Ann. Sc. E. N. S., 4?me s?rie, t.15, (1982), 213-230 · Zbl 0501.53030
[4] Carlen E., Kusuoka S., Stroock D: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincar?, Prob. Stat.,23 (1987), 245-287 · Zbl 0634.60066
[5] Chavel I.: Eigenvalues in Riemannian Geometry. Academic Press, (1984) · Zbl 0551.53001
[6] Cheeger J., Gromov M., Taylor M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Diff. Geo.17 (1982), 15-53 · Zbl 0493.53035
[7] Coulhon T.: Espaces de Lipschitz et in?galit?s de Poincar?. (1994)
[8] Coulhon T., Saloff-Coste L.: Isop?rim?trie sur les groupes et les vari?t?s. Rev. Mat. Iberoamericana,9 (1993), 293-314 · Zbl 0782.53066
[9] Coulhon T., Saloff-Coste L.: Vari?t?s riemanniennes isom?triques ? l’infini, Rev. Mat. Iber. para?tre · Zbl 0845.58054
[10] Croke C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ec. Norm. Sup. Paris,13 (1980), 419-435 · Zbl 0465.53032
[11] Davies E.B.: Heat kernels and spectral theory. Cambridge U.P., (1989) · Zbl 0699.35006
[12] Fefferman C., Phong D.H.: Subelliptic eigenvalue problems. In Proc. Conf. Harm. Anal. in Honor of A. Zygmund, Wadsworth Math. Ser. Wadsworth, Belmont, California (1981), 590-606 · Zbl 0503.35071
[13] Franchi B., Gallot S., Wheeden R.L.: Sobolev and isoperimetric inequalities for degenerate metrics. Math. Ann., para?tre · Zbl 0830.46027
[14] Franchi B., Gutierrez C., Wheeden R.L.: Weighted Sobolev-Poincar? inequalities for Gruschin type operators. Comm. P.D.E.,19 (1994), 523-604. · Zbl 0822.46032 · doi:10.1080/03605309408821025
[15] Fukushima.: Dirichlet forms and Markov processes. North Holland, Amsterdam (1980) · Zbl 0422.31007
[16] Guivarc’h Y.: Croissance polyn?miale et p?riodes des fonctions harmoniques. Bull. Soc. Math. France,101 (1973), 333-379
[17] Jerison D.: The Poincar? inequality for vector fields satisfying H?rmander condition. Duke Math. J.,53 (1986), 503-523 · Zbl 0614.35066 · doi:10.1215/S0012-7094-86-05329-9
[18] Jerison D., Sanchez-Calle A.: Subelliptic second order differential operators. Springer Lecture Notes in Math.,1277 (1986), 47-77
[19] Kusuoka S., Stroock D.: Applications of the Malliavin calculus, III., J. Fac. Sci. Univ. Tokyo, Sect. IA, Math.,34 (1987), 391-442 · Zbl 0633.60078
[20] Lu G.: Weighted Poincar? and Sobolev inequalities for vector fields satisfying H?rmander condition. Rev. Mat. Iber.,8 (1992), 367-439 · Zbl 0804.35015
[21] Lu G.: The sharp Poincar? inequality for free vector fields: An end point result. Rev. Mat. Iber.,10 (1994), 1-14
[22] Maheux P.: Analyse et g?om?trie sur les espaces homog?nes. Th?se de Doctorat, Universit? Paris VI (1991)
[23] Maheux P.: Estimations du noyau de la chaleur sur les espaces homog?nes. A para?tre J. Geom. Anal. · Zbl 1044.58029
[24] Maz’ya V.: Sobolev spaces. Springer Verlag, (1985)
[25] Nagel A., Stein E., Wainger M.: Balls and metrics defined by vector fields. Acta Math.,155 (1985), 103-147 · Zbl 0578.32044 · doi:10.1007/BF02392539
[26] Robinson D.: Elliptic operators on Lie groups. Oxford University Press, (1991) · Zbl 0764.47025
[27] Saloff-Coste L., Stroock D.: Op?rateurs uniform?ment sous-elliptiques sur des groupes de Lie. J. Funct. Anal.,98 (1991), 97-109 · Zbl 0734.58041 · doi:10.1016/0022-1236(91)90092-J
[28] Saloff-Coste L.: Analyse sur les groupes de Lie ? croissance polyn?miale. Arkiv f?r Mat.,28 (1990), 315-331 · Zbl 0715.43009 · doi:10.1007/BF02387385
[29] Saloff-Coste L.: Uniformly elliptic operators on Riemannian manifolds. J. Diff. Geom.,36 (1992), 417-450 · Zbl 0735.58032
[30] Saloff-Coste L.: A note on Poincar?, Sobolev, and Harnack inequalities. Duke Math. J., I.M.R.N.,2 (1992), 27-38 · Zbl 0769.58054
[31] Saloff-Coste L.: On global Sobolev inequalities. Forum Math.,67 (1994), 109-121 · Zbl 0816.53027
[32] Sturm K. Th.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness andL p Liouville properties. J. Reine Angew. Math., (1994)
[33] Varopoulos N., Analysis on Lie groups. J. funct. Anal.,76 (1988), 346-410 · Zbl 0634.22008 · doi:10.1016/0022-1236(88)90041-9
[34] Varopoulos N.: Fonctions harmoniques sur les groupes de Lie. C.R.A.S. Paris, s?rie I,304 (1987), 519-521 · Zbl 0614.22002
[35] Varopoulos N.: Small time Gaussian estimates of heat diffusion kernels. Part I: the semigroup technique. Bull. Sc. Math.,113 (1989), 253-277 · Zbl 0703.58052
[36] Varopoulos N., Saloff-Cost L., Coulhon T.: Analysis and geometry on groups. Cambridge University Press, (1993)
[37] Yau S-T.: Isoperimetric constants and the first eigenvalue of a compact riemannian manifold. Ann. Sc. E.N.S. 4?me s?rie,8 (1975), 487-507 · Zbl 0325.53039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.