Duke Mathematical Journal

Sharp inequalities for functional integrals and traces of conformally invariant operators

Carlo Morpurgo

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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

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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. http://projecteuclid.org/euclid.dmj/1087575456.

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