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15 September 2002 Sharp inequalities for functional integrals and traces of conformally invariant operators
Carlo Morpurgo
Duke Math. J. 114(3): 477-553 (15 September 2002). DOI: 10.1215/S0012-7094-02-11433-1

Abstract

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.

Citation

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Carlo Morpurgo. "Sharp inequalities for functional integrals and traces of conformally invariant operators." Duke Math. J. 114 (3) 477 - 553, 15 September 2002. https://doi.org/10.1215/S0012-7094-02-11433-1

Information

Published: 15 September 2002
First available in Project Euclid: 18 June 2004

zbMATH: 1065.58022
MathSciNet: MR1 924 571
Digital Object Identifier: 10.1215/S0012-7094-02-11433-1

Subjects:
Primary: 58J40
Secondary: 35B05 , 35P20 , 47G30 , 60J25

Rights: Copyright © 2002 Duke University Press

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Vol.114 • No. 3 • 15 September 2002
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