Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere. (English) Zbl 1334.35366

Branson, Thomas P.; Fontana, Luigi; Morpurgo, Carlo

Ann. Math. (2) 177, No. 1, 1-52 (2013).

In this paper, the authors obtain three related results. First, they study Moser-Trudinger inequalities on the CR sphere and obtain sharp estimates. Then, they present a version of the Beckner-Onofri inequality on the space of CR-plurisubharmonic functions. Finally, they investigate the Hardy-Littlewood-Sobolev inequality as a consequence of the second result.

Reviewer: Yunus E. Zeytuncu (Dearborn)

MSC:

35R01	PDEs on manifolds
35R03	PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.

Keywords:

Beckner-Onofri; conformal invariance; CR sphere; Heisenberg group; intertwining operators; Moser-Trudinger; sharp inequalities

References:

[1]	D. R. Adams, ”A sharp inequality of J. Moser for higher order derivatives,” Ann. of Math., vol. 128, iss. 2, pp. 385-398, 1988. · Zbl 0672.31008
[2]	F. Astengo, M. Cowling, and B. Di Blasio, ”The Cayley transform and uniformly bounded representations,” J. Funct. Anal., vol. 213, iss. 2, pp. 241-269, 2004. · Zbl 1054.22006
[3]	F. Astengo and B. Di Blasio, ”Sobolev spaces and the Cayley transform,” Proc. Amer. Math. Soc., vol. 134, iss. 5, pp. 1319-1329, 2006. · Zbl 1091.43003
[4]	T. Aubin, ”Problèmes isopérimétriques et espaces de Sobolev,” J. Differential Geometry, vol. 11, iss. 4, pp. 573-598, 1976. · Zbl 0371.46011
[5]	T. Aubin, ”Meilleures constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire,” J. Funct. Anal., vol. 32, iss. 2, pp. 148-174, 1979. · Zbl 0411.46019
[6]	Z. M. Balogh, J. J. Manfredi, and J. T. Tyson, ”Fundamental solution for the \(Q\)-Laplacian and sharp Moser-Trudinger inequality in Carnot groups,” J. Funct. Anal., vol. 204, iss. 1, pp. 35-49, 2003. · Zbl 1080.22003
[7]	C. Bandle, Isoperimetric Inequalities and Applications, Boston, MA: Pitman (Advanced Publishing Program), 1980, vol. 7. · Zbl 0436.35063
[8]	W. Beckner, ”Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,” Ann. of Math., vol. 138, iss. 1, pp. 213-242, 1993. · Zbl 0826.58042
[9]	C. Benson, A. H. Dooley, and G. Ratcliff, ”Fundamental solutions for powers of the Heisenberg sub-Laplacian,” Illinois J. Math., vol. 37, iss. 3, pp. 455-476, 1993. · Zbl 0789.22012
[10]	T. P. Branson, ”Sharp inequalities, the functional determinant, and the complementary series,” Trans. Amer. Math. Soc., vol. 347, iss. 10, pp. 3671-3742, 1995. · Zbl 0848.58047
[11]	T. P. Branson, Memo to Noël Lohoué, 1999.
[12]	T. P. Branson, S. A. Chang, and P. C. Yang, ”Estimates and extremals for zeta function determinants on four-manifolds,” Comm. Math. Phys., vol. 149, iss. 2, pp. 241-262, 1992. · Zbl 0761.58053
[13]	T. P. Branson, L. Fontana, and C. Morpurgo, Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere, 2007. · Zbl 1334.35366
[14]	T. P. Branson, G. Ólafsson, and B. Ørsted, ”Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup,” J. Funct. Anal., vol. 135, iss. 1, pp. 163-205, 1996. · Zbl 0841.22011
[15]	E. Carlen and M. Loss, ”Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on \(S^n\),” Geom. Funct. Anal., vol. 2, iss. 1, pp. 90-104, 1992. · Zbl 0754.47041
[16]	D. Chang and J. Tie, ”Estimates for powers of sub-Laplacian on the non-isotropic Heisenberg group,” J. Geom. Anal., vol. 10, iss. 4, pp. 653-678, 2000. · Zbl 0992.22007
[17]	S. A. Chang and J. Qing, ”The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set,” J. Funct. Anal., vol. 147, iss. 2, pp. 363-399, 1997. · Zbl 0914.58040
[18]	S. A. Chang and P. C. Yang, ”Extremal metrics of zeta function determinants on \(4\)-manifolds,” Ann. of Math., vol. 142, iss. 1, pp. 171-212, 1995. · Zbl 0842.58011
[19]	S. A. Chang and P. C. Yang, ”Prescribing Gaussian curvature on \(S^2\),” Acta Math., vol. 159, iss. 3-4, pp. 215-259, 1987. · Zbl 0636.53053
[20]	S. Chanillo, H. -L. Chiu, and P. Yang, ”Embeddability for Three-Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants,” Duke Math. J., vol. 161, 2012. · Zbl 1271.32040
[21]	W. S. Cohn and G. Lu, ”Best constants for Moser-Trudinger inequalities on the Heisenberg group,” Indiana Univ. Math. J., vol. 50, iss. 4, pp. 1567-1591, 2001. · Zbl 1019.43009
[22]	W. S. Cohn and G. Lu, ”Sharp constants for Moser-Trudinger inequalities on spheres in complex space \(\mathbb C^n\),” Comm. Pure Appl. Math., vol. 57, iss. 11, pp. 1458-1493, 2004. · Zbl 1063.35060
[23]	M. Cowling, ”Unitary and uniformly bounded representations of some simple Lie groups,” in Harmonic Analysis and Group Representations, Naples: Liguori, 1982, pp. 49-128.
[24]	D. G. de Figueiredo, Lectures on the Ekeland variational principle with applications and detours, Published for the Tata Institute of Fundamental Research, Bombay, 1989. · Zbl 0688.49011
[25]	G. David, J. -L. Journé, and S. Semmes, ”Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation,” Rev. Mat. Iberoamericana, vol. 1, iss. 4, pp. 1-56, 1985. · Zbl 0604.42014
[26]	M. Demuth and J. A. van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators, Basel: Birkhäuser, 2000. · Zbl 0980.60005
[27]	C. Fefferman and K. Hirachi, ”Ambient metric construction of \(Q\)-curvature in conformal and CR geometries,” Math. Res. Lett., vol. 10, iss. 5-6, pp. 819-831, 2003. · Zbl 1166.53309
[28]	G. B. Folland, ”A fundamental solution for a subelliptic operator,” Bull. Amer. Math. Soc., vol. 79, pp. 373-376, 1973. · Zbl 0256.35020
[29]	G. B. Folland, ”Subelliptic estimates and function spaces on nilpotent Lie groups,” Ark. Mat., vol. 13, iss. 2, pp. 161-207, 1975. · Zbl 0312.35026
[30]	L. Fontana, ”Sharp borderline Sobolev inequalities on compact Riemannian manifolds,” Comment. Math. Helv., vol. 68, iss. 3, pp. 415-454, 1993. · Zbl 0844.58082
[31]	L. Fontana and C. Morpurgo, ”Adams inequalities on measure spaces,” Adv. Math., vol. 226, iss. 6, pp. 5066-5119, 2011. · Zbl 1219.46032
[32]	R. L. Frank and E. H. Lieb, ”Sharp constants in several inequalities on the Heisenberg group,” Ann. of Math., vol. 176, pp. 349-381, 2012. · Zbl 1252.42023
[33]	D. Geller, ”The Laplacian and the Kohn Laplacian for the sphere,” J. Differential Geom., vol. 15, iss. 3, pp. 417-435 (1981), 1980. · Zbl 0507.58049
[34]	R. A. Gover and R. C. Graham, ”CR invariant powers of the sub-Laplacian,” J. Reine Angew. Math., vol. 583, pp. 1-27, 2005. · Zbl 1076.53048
[35]	R. C. Graham, ”Compatibility operators for degenerate elliptic equations on the ball and Heisenberg group,” Math. Z., vol. 187, iss. 3, pp. 289-304, 1984. · Zbl 0538.47031
[36]	R. C. Graham, R. Jenne, L. Mason, and G. Sparling, ”Conformally invariant powers of the Laplacian. I. Existence,” J. London Math. Soc., vol. 46, iss. 3, pp. 557-565, 1992. · Zbl 0726.53010
[37]	J. Hersch, ”Quatre propriétés isopérimétriques de membranes sphériques homogènes,” C. R. Acad. Sci. Paris Sér. A-B, vol. 270, p. a1645-a1648, 1970. · Zbl 0224.73083
[38]	K. Hirachi, ”Scalar pseudo-Hermitian invariants and the Szeg\Ho kernel on three-dimensional CR manifolds,” in Complex Geometry, New York: Dekker, 1993, vol. 143, pp. 67-76. · Zbl 0805.32014
[39]	D. Jerison and J. M. Lee, ”The Yamabe problem on CR manifolds,” J. Differential Geom., vol. 25, iss. 2, pp. 167-197, 1987. · Zbl 0661.32026
[40]	D. Jerison and J. M. Lee, ”Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem,” J. Amer. Math. Soc., vol. 1, iss. 1, pp. 1-13, 1988. · Zbl 0634.32016
[41]	K. D. Johnson and N. R. Wallach, ”Composition series and intertwining operators for the spherical principal series. I,” Trans. Amer. Math. Soc., vol. 229, pp. 137-173, 1977. · Zbl 0349.43010
[42]	E. H. Lieb, ”Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,” Ann. of Math., vol. 118, iss. 2, pp. 349-374, 1983. · Zbl 0527.42011
[43]	A. Korányi and H. M. Reimann, ”Quasiconformal mappings on the Heisenberg group,” Invent. Math., vol. 80, iss. 2, pp. 309-338, 1985. · Zbl 0567.30017
[44]	A. Korányi and H. M. Reimann, ”Foundations for the theory of quasiconformal mappings on the Heisenberg group,” Adv. Math., vol. 111, iss. 1, pp. 1-87, 1995. · Zbl 0876.30019
[45]	C. Morpurgo, ”The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions on \(S^n\),” Geom. Funct. Anal., vol. 6, iss. 1, pp. 146-171, 1996. · Zbl 0852.58079
[46]	C. Morpurgo, ”Sharp inequalities for functional integrals and traces of conformally invariant operators,” Duke Math. J., vol. 114, iss. 3, pp. 477-553, 2002. · Zbl 1065.58022
[47]	J. Moser, ”A sharp form of an inequality by N. Trudinger,” Indiana Univ. Math. J., vol. 20, pp. 1077-1092, 1970/71. · Zbl 0213.13001
[48]	J. Moser, ”On a nonlinear problem in differential geometry,” in Dynamical Systems, New York: Academic Press, 1973, pp. 273-280. · Zbl 0275.53027
[49]	K. Okikiolu, ”Extremals for logarithmic Hardy-Littlewood-Sobolev inequalities on compact manifolds,” Geom. Funct. Anal., vol. 17, iss. 5, pp. 1655-1684, 2008. · Zbl 1140.58003
[50]	E. Onofri, ”On the positivity of the effective action in a theory of random surfaces,” Comm. Math. Phys., vol. 86, iss. 3, pp. 321-326, 1982. · Zbl 0506.47031
[51]	B. Osgood, R. Phillips, and P. Sarnak, ”Extremals of determinants of Laplacians,” J. Funct. Anal., vol. 80, iss. 1, pp. 148-211, 1988. · Zbl 0653.53022
[52]	R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, London: Pitman, 1977, vol. 1. · Zbl 0364.35001
[53]	N. K. Stanton, ”Spectral invariants of CR manifolds,” Michigan Math. J., vol. 36, iss. 2, pp. 267-288, 1989. · Zbl 0685.58033
[54]	G. Talenti, ”Best constant in Sobolev inequality,” Ann. Mat. Pura Appl., vol. 110, pp. 353-372, 1976. · Zbl 0353.46018
[55]	N. S. Trudinger, ”On imbeddings into Orlicz spaces and some applications,” J. Math. Mech., vol. 17, pp. 473-483, 1967. · Zbl 0163.36402
[56]	J. N. Vilenkin and A. U. Klimyk, Representation of Lie groups and Special Functions. Vol. 2. Class I Representations, Special Functions, and Integral Transforms, Dordrecht: Kluwer Academic Publishers Group, 1993, vol. 74. · Zbl 0809.22001
[57]	K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, New York: Springer-Verlag, 2005, vol. 226. · Zbl 1067.32005