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Singularities, Supports and Lacunas

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Advances in Microlocal Analysis

Part of the book series: NATO ASI Series ((ASIC,volume 168))

Abstract

These notes are a survey of three aspects of the modern theory of linear partial differential equations, and its generalization to the microlocal analysis of pseudodifferential operators. The first chapter is a study of the propagation of singularities of partial and pseudo differential equations, beginning with a sketch of the extensive background of pseudo differential and Fourier integral operators and wave front sets — the machinery of microlocal analysis in phase space. Selected results on equations with multiple characteristics are then discussed, in the involutive and non-involutive cases. The second chapter is a description of the work of C. Fefferman and others on the approximate simultaneous diagonalization of differential operators with variable coefficients, regarded as algebraic operators in phase space. The uncertainty principle, a title borrowed from Heisenberg’s quantum mechanics, limits the precision of this process, since a function and its Fourier transform cannot both have small supports. This area of investigation, which draws upon the full resources of microlocal analysis, appears to have interesting future prospects.

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  1. Department of Mathematics, University of Toronto, Toronto, Canada, M5S 1A1

    G. F. D. Duff

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  1. Inst. de Math., University of Liège, 15, avenue des Tilleuls, B-4000, Liège, Belgium

    H. G. Garnir

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Duff, G.F.D. (1986). Singularities, Supports and Lacunas. In: Garnir, H.G. (eds) Advances in Microlocal Analysis. NATO ASI Series, vol 168. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4606-4_4

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