Abstract
The functional estimated sharply by the logarithmic Hardy-Littlewood-Sobolev inequality onS n, for evenn≥2, is linked to the spectrum of the Paneitz operators. From the point of view of conformal geometry such operators are naturaln-dimensional generalizations of the Laplacian onS 2.
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References
T. Aubin, Nonlinear Analysis on Manifolds, Monge-Ampère Equations, Springer-Verlag, New York, 1982.
W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. 138 (1993), 213–242.
M. Berger, P. Gauduchon, E. Mazet, Le spectre d'une variété riemannienne, Springer Lecture Notes in Math. 194 (1971).
T.P. Branson, Sharp inequalities, the functional determinant and the complementary series, to appear in Trans. Amer. Math. Soc.
T.P. Branson, Differential operators canonically associated with a conformal structure, Math. Scand. 57 (1985), 293–345.
T.P. Branson, An anomaly associated with 4-dimensional quantum gravity, to appear in Commun. Math. Phys.
T.P. Branson, S.Y.A. Chang, P. Yang, Estimates and extremals for zeta function determinants on four-manifolds, Commun. Math. Phys. 149 (1992), 241–262.
T.P. Branson, B. Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), 261–293.
T.P. Branson, B. Ørsted, Explicit functional determinants in four dimensions, Proc. Am. Math. Soc. 113 (1991), 669–682.
T.P. Branson, B. Ørsted, Conformal geometry and global invariants, Diff. Geom. Appl. 1 (1991), 279–308.
E. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality onS n, GAFA 2 (1992), 90–104.
Chang S.-Y. A., P. Yang, Extremal metrics of zeta function determinants on 4-manifolds, to appear in Annals of Math.
P. Chiu, Extremal Determinants, Dense Sphere Packings, and Covering with Hecke Points, Ph.D. dissertation, Stanford University, 1991.
E.B. Davies, Long time asymptotics of fourth order parabolic equations, preprint, 1994.
E.B. Davies, Uniformly elliptic operators with measurable coefficients, to appear in J. Funct. Anal.
E.B. Davies, Pointwise bounds on the space and time derivatives of heat kernels, J. Oper. Theory 21 (1989), 367–378.
P. Gilkey, Invariance Theory, the Heat Equation, and the Atyiah-Singer Index Theorem, CRC Press, Boca Raton, 1995.
P. Gilkey, The spectral geometry of the higher order Laplacian, Duke Math. J. 47 (1980), 511–528.
C.R. Graham, R. Jenne, L. Mason, G. Sparling, Conformally invariant powers of the Laplacian, I: existence, J. London Math. Soc. 46 (1992), 557–565.
P. Greiner, An asymptotic expansion for the heat kernel, Arch. Rat. Mech. Anal. 41 (1979), 163–218.
J. Hersch, Quatre propriétés isopérimétrique de membranes sphérique, C.R. Acad. Sci., Paris, 270 (1970), 1645–1648.
E.H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. 118 (1983), 349–374.
C. Morpurgo, Local extrema of traces of heat kernels onS 2, to appear in J. Funct. Anal.
C. Morpurgo, Conformal perturbation of heat kernels and their traces, to appear in J. Funct. Anal.
E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys. 86 (1982), 321–326.
B. Osgood, R. Phillips, P. Sarnak, Extremals of determinants of laplacians, J. Funct. Anal. 80 (1988), 148–211.
S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint 1983.
T. Parker, S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geom. 25 (1987), 199–222.
C. Sogge, Fourier Integrals in Classical Analysis, Cambridge Univ. Press, 1993.
A. Terras, Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Amer. Math. Soc. 183 (1973), 477–486.
E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, Oxford, 1951.
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Morpurgo, C. The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions onS n . Geometric and Functional Analysis 6, 146–171 (1996). https://doi.org/10.1007/BF02246771
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DOI: https://doi.org/10.1007/BF02246771