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Analysis on Root Systems

Published online by Cambridge University Press:  20 November 2018

Amédée Debiard  and
Bernard Gaveau
Amédée Debiard
Affiliation:
Université Paris XIII, Paris, France
Bernard Gaveau
Affiliation:
Université Paris VI, Paris, France
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A great part of mathematical analysis relies directly on the methods of separation of variables and on the successive reduction of several variables problems to one-dimensional equations and to the theory of classical special functions; for example, the theory of elliptic or parabolic equations with regular coefficients (even with non constant coefficients) can be done because we know explicitly the fundamental solutions of the Laplace operator or of the heat equation; these fundamental solutions are functions of one variable; pseudodifferential or parametrices methods are thus basically small perturbations of an explicitly known problem in one variable.

On the other hand, there are many problems which are not of this type: they are related to the questions of operators with singular coefficients and to the global behaviour of the solutions; in that case, the local model cannot be reduced to a one variable problem but is fundamentally a several variables problem which cannot be treated in a detailed way by one variable methods or perturbation analysis of a one variable problem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 6 , 01 December 1987 , pp. 1281 - 1404
Copyright
Copyright © Canadian Mathematical Society 1987

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