An interpolation of Hardy inequality and Moser-Trudinger inequality on Riemannian manifolds with negative curvature. (English) Zbl 1361.46027
Summary: Let \(M\) be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of the Hardy inequality and the Moser-Trudinger inequality on \(M\). Furthermore, the constant we obtain is sharp.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 58E35 | Variational inequalities (global problems) in infinite-dimensional spaces |
References:
| [1] | Adams, D. R.: A sharp inequality of J. Moser for higher order derivatives. Ann. of Math., 128, 385-398 (1988) · Zbl 0672.31008 · doi:10.2307/1971445 |
| [2] | Adimurthi, Yang, Y.: An interpolation of Hardy inequality and Trudinger-Moser inequality in Rn and its applications. Int. Math. Res. Not., 13, 2394-2426 (2010) · Zbl 1198.35012 |
| [3] | Aubin, T.: Sur la function exponentielle. C. R. Math. Acad. Sci. Paris, 270, 1514 (1970) · Zbl 0197.47802 |
| [4] | Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. of Math., 138, 213-242 (1993) · Zbl 0826.58042 · doi:10.2307/2946638 |
| [5] | Carron, G.: Inégalités de Hardy sur les variétés riemanniennes non-compactes. J. Math. Pures Appl., 76, 883-891 (1997) · Zbl 0886.58111 · doi:10.1016/S0021-7824(97)89976-X |
| [6] | Cherrier, P.: Une inégalité de Sobolev sur les variétés Riemanniennes. Bull. Sci. Math., 103, 353-374 (1979) · Zbl 0414.35074 |
| [7] | Cherrier, P.: Cas déxception du théorème d’inclusion deSobolev sur les variétés Riemanniennes et applications. Bull. Sci. Math., 105, 235-288 (1981) · Zbl 0471.58026 |
| [8] | Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Comment. Math. Helv., 68, 415-454 (1993) · Zbl 0844.58082 · doi:10.1007/BF02565828 |
| [9] | Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, Third ed, Springer-Verlag, Berlin, 2004 · Zbl 1068.53001 · doi:10.1007/978-3-642-18855-8 |
| [10] | Kombe, I., Özaydin, M.: Improved Hardy and Rellich inequalities on Riemannian manifolds. Trans. Amer. Math. Soc., 361, 6191-6203 (2009) · Zbl 1178.26013 · doi:10.1090/S0002-9947-09-04642-X |
| [11] | Lam, N., Lu, G.: The sharp singular Adams inequalities in high order Sobolev spaces, arXiv:1112.6431v1 · Zbl 1319.46027 |
| [12] | Lam, N., Lu, G.: Sharp Moser-Trudinger inequality in the Heisenberg group at the critical case and applications. Adv. Math., 231, 3259-3287 (2012) · Zbl 1278.42033 · doi:10.1016/j.aim.2012.09.004 |
| [13] | Lam, N., Lu, G.: A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangementfree argumnet. J. Differential Equations, 255, 298-325 (2013) · Zbl 1294.46034 · doi:10.1016/j.jde.2013.04.005 |
| [14] | Li, J., Lu, G.: Critical and subcritical Moser-Trudinger inequalities on complete noncompact Riemannian Manifolds, Preprint · Zbl 1482.46040 |
| [15] | Li, P., Lecture Notes on Geometric Analysis, Lecture Notes Series (1993), Seoul · Zbl 0822.58001 |
| [16] | Li, Y. X., Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in Rn. Indiana Univ. Math. J., 57, 451-480 (2008) · Zbl 1157.35032 · doi:10.1512/iumj.2008.57.3137 |
| [17] | Lu, G., Tang, H.: Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces. Adv. Nonlinear Stud., 13, 1035-1052 (2013) · Zbl 1294.46035 |
| [18] | Lu, G., Tang, H.: Sharp Moser-Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition. J. Geom. Anal., 26, 837-857 (2016) · Zbl 1356.46031 · doi:10.1007/s12220-015-9573-y |
| [19] | Lu, G., Yang, Q.: A sharp Trudinger-Moser inequality on any bounded and convex planar domain, arXiv:1512.07163 · Zbl 1369.46029 |
| [20] | Mancini, G., Sandeep, K.: Moser-Trudinger inequality on conformal discs. Commun. Contemp. Math., 12, 1055-1068 (2010) · Zbl 1205.35067 · doi:10.1142/S0219199710004111 |
| [21] | Mancini, G., Sandeep, K., Tintarev K.: Trudinger-Moser inequality in the hyperbolic space HN. Adv. Nonlinear Stud., 2, 309-324 (2013) · Zbl 1274.35435 |
| [22] | Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J., 20, 1077-1092 (1970) · Zbl 0203.43701 · doi:10.1512/iumj.1971.20.20101 |
| [23] | Ruf, B.: A sharp Trudinger-Moser type inequality for unbounded domains in R2. J. Funct. Anal., 219, 340-367 (2005) · Zbl 1119.46033 · doi:10.1016/j.jfa.2004.06.013 |
| [24] | Ruf, B., Sani, F.: Sharp Adams-type inequalities in Rn. Trans. Amer. Math. Soc., 365, 645-670 (2013) · Zbl 1280.46024 · doi:10.1090/S0002-9947-2012-05561-9 |
| [25] | Schoen, R., Yau, S.-T.: Lectures on Differential Geometry, Vol. 1, International Press, Somerville, Massachusetts, 1994 · Zbl 0830.53001 |
| [26] | Trudinger, N. S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech., 17, 473-483 (1967) · Zbl 0163.36402 |
| [27] | Yang, Y.: Trudinger-Moser inequalities on complete noncompact Riemannian manifolds. J. Funct. Anal., 263, 1894-1938 (2012) · Zbl 1256.53034 · doi:10.1016/j.jfa.2012.06.019 |
| [28] | Yang, Q., Su, D., Kong, Y.: Hardy inequalities on Riemannian manifolds with negative curvature. Commun. Contemp. Math., 16, 1-24 (2014) · Zbl 1294.26019 · doi:10.1090/conm/622/12430 |
| [29] | Yang, Q., Su, D., Kong, Y.: Sharp Moser-Trudinger inequalities on Riemannian manifolds with Negative curvature. Ann. Mat. Pura Appl., 195, 459-471 (2016) · Zbl 1355.46044 · doi:10.1007/s10231-015-0472-4 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
