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The problem of lacunas and analysis on root systems


Author: Yuri Berest
Journal: Trans. Amer. Math. Soc. 352 (2000), 3743-3776
MSC (1991): Primary 58F07, 35L15; Secondary 58F37, 35L25
Published electronically: April 18, 2000
MathSciNet review: 1694280
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Abstract:

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.


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Additional Information

Yuri Berest
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Email: berest@math.cornell.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02543-5
Keywords: Hyperbolic linear differential operators, fundamental solution, lacuna, Huygens' principle, Coxeter groups, Dunkl operators
Received by editor(s): March 6, 1997
Received by editor(s) in revised form: November 3, 1997
Published electronically: April 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society