Duke Mathematical Journal

Sharp inequalities for functional integrals and traces of conformally invariant operators

Carlo Morpurgo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The intertwining operators $A\sb d=A\sb d(g)$ on the round sphere $(S\sp n,g)$ are the conformal analogues of the power Laplacians $\Delta\sp {d/2}$ on the flat $\mathbf {R}\sp n$. To each metric $\rho g$, conformally equivalent to $g$, we can naturally associate an operator $A\sb d(\rho g)$, which is compact, elliptic, pseudodifferential of order $d$, and which has eigenvalues $\lambda\sb j(\rho)$; the special case $d=2$ gives precisely the conformal Laplacian in the metric $\rho g$. In this paper we derive sharp inequalities for a class of trace functionals associated to such operators, including their zeta function $\sum\sp j\lambda\sp j(\rho)\sp {-s}$, and its regularization between the first two poles. These inequalities are expressed analytically as sharp, conformally invariant Sobolev-type (or log Sobolev type) inequalities that involve either multilinear integrals or functional integrals with respect to $d$-symmetric stable processes. New strict rearrangement inequalities are derived for a general class of path integrals.

Article information

Duke Math. J., Volume 114, Number 3 (2002), 477-553.

First available in Project Euclid: 18 June 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)
MR1 924 571

Zentralblatt MATH identifier

Primary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx] 60J25: Continuous-time Markov processes on general state spaces


Morpurgo, Carlo. Sharp inequalities for functional integrals and traces of conformally invariant operators. Duke Math. J. 114 (2002), no. 3, 477--553. doi:10.1215/S0012-7094-02-11433-1. https://projecteuclid.org/euclid.dmj/1087575456

Export citation


  • W. Arendt and C. J. K. Batty, Absorption semigroups and Dirichlet boundary conditions, Math. Ann. 295 (1993), 427--448.
  • A. Baernstein II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and *-functions in $n$-space, Duke Math. J. 43 (1976), 245--268.
  • W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), 4816--4819.
  • --. --. --. --., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), 213--242.
  • --. --. --. --., ``Geometric inequalities in Fourier analysis'' in Essays on Fourier Analysis in Honor of Elias Stein (Princeton, 1991), Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, 1995, 36--68.
  • --. --. --. --., Logarithmic Sobolev inequalities and the existence of singular integrals, Forum Math. 9 (1997), 303--323.
  • R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263--273.
  • T. P. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995), 3671--3742.
  • T. P. Branson, S.-Y. A. Chang, and P. C. Yang, Estimates and extremals for zeta function determinants on four-manifolds, Comm. Math. Phys. 149 (1992), 241--262.
  • H. J. Brascamp, E. H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227--237.
  • A. Burchard and M. Schmuckenschläger, Comparison theorems for exit times, Geom. Funct. Anal. 11 (2001), 651--692.
  • E. Carlen and M. Loss, Extremals of functionals with competing symmetries, J. Funct. Anal. 88 (1990), 437--456.
  • --. --. --. --., Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $S^n$, Geom. Funct. Anal. 2 (1992), 90--104.
  • S.-Y. A. Chang and J. Qing, The zeta functional determinants on manifolds with boundary, II: Extremal metrics and compactness of isospectral set, J. Funct. Anal. 147 (1997), 363--399.
  • S.-Y. A. Chang and P. Yang, Extremal metrics of zeta function determinants on $4$-manifolds, Ann. of Math.(2) 142 (1995), 171--212.
  • Z.-Q. Chen and R. Song, Intrinsic ultracontractivity and conditional gauge for symmetric stable processes, J. Funct. Anal. 150 (1997), 204--239.
  • K. L. Chung, Lectures from Markov Processes to Brownian Motion, Grundlehren Math. Wiss. 249, Springer, New York, 1982.
  • M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. (2) 106 (1977), 93--100.
  • I. Daubechies, An uncertainty principle for fermions with generalized kinetic energy, Comm. Math. Phys. 90 (1983), 511--520.
  • M. Demuth and J. M. van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration Approach, Probab. Appl., Birkhäuser, Basel, 2000.
  • A. Erdélyi, W. Mangus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. I, based in part on notes left by H. Bateman, McGraw Hill, New York, 1953.
  • S. N. Ethier and T. G. Kurtz, Markov Processes: A Functional Integration Approach, Wiley, New York, 1986.
  • R. Friedberg and J. M. Luttinger, Rearrangement inequality for periodic functions, Arch. Rational Mech. Anal. 61 (1976), 35--44.
  • F. Gesztesy and Z. Zhao, Domain perturbations, Brownian motion, capacities, and ground states of Dirichlet Schrödinger operators, Math. Z. 215 (1994), 143--150.
  • C. R. Graham, R. Jenne, L. Mason, and G. Sparling, Conformally invariant powers of the Laplacian, I: Existence, J. London Math. Soc. (2) 46 (1992), 557--565.
  • P. Greiner, An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal. 41 (1971), 163--218.
  • G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, 2d ed., Progr. Math. 65, Birkhäuser, Boston, 1996.
  • I. W. Herbst and Z. Zhao, ``Sobolev spaces, Kac-regularity, and the Feynmanc-Kac formula'' in Seminar on Stochastic Processes (Princeton, 1987), Progr. Probab. Statist. 15, Birkhäuser, Boston, 1988, 171--191.
  • W. Hoh and N. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Funct. Anal. 137 (1996), 19--48.
  • E. H. Lieb, Bounds on the eigenvalues of the Laplace and Schroedinger operators, Bull. Amer. Math. Soc. 82 (1976), 751--753.
  • --. --. --. --., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349--374.
  • E. H. Lieb and M. Loss, Analysis, Grad. Stud. Math. 14, Amer. Math. Soc., Providence, 1997.
  • J. M. Luttinger, Generalized isoperimetric inequalities, J. Math. Phys. 14 (1973), 586--593.
  • D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Math. Appl. (East European Ser.) 61, Kluwer, Dordrecht, 1993.
  • C. Morpurgo, The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions on $S^n$, Geom. Funct. Anal. 6 (1996), 146--171.
  • K. Okikiolu, Critical metrics for the determinant of the Laplacian in odd dimensions, Ann. of Math. (2) 153 (2001), 471--531. \CMP1 829 756
  • --------, Critical metrics for spectral zeta functions, preprint, 2001.
  • H. Ôkura, On the spectral distributions of certain integrodifferential operators with random potential, Osaka J. Math. 16 (1979), 633--666.
  • F. W. J. Olver, Asymptotics and Special Functions, Comput. Sci. Appl. Math., Academic Press, New York, 1974.
  • 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, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148--211.
  • L. Peterson, Conformally covariant pseudo-differential operators, Differential Geom. Appl. 13 (2000), no. 2, 197--211.
  • S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Probab. Math. Statist., Academic Press, New York, 1978.
  • R. T. Seeley, ``Complex powers of an elliptic operator'' in Singular Integrals (Chicago, 1966), Proc. Sympos. Pure Math. 10, Amer. Math. Soc., Providence, 1967, 288--307.
  • B. Simon, Functional Integration and Quantum Physics, Pure Appl. Math. 86, Academic Press, New York, 1979.
  • D. Stroock, The Kac Approach to potential theory, I, J. Math. Mech. 16 (1967), 829--852.