On local solvability of linear partial differential equations—part II. Sufficient conditions†
01 May 1970-Communications on Pure and Applied Mathematics (Wiley Subscription Services, Inc., A Wiley Company)-Vol. 23, Iss: 3, pp 459-509
About: This article is published in Communications on Pure and Applied Mathematics.The article was published on 1970-05-01. It has received 122 citations till now. The article focuses on the topics: First-order partial differential equation & Parabolic partial differential equation.
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TL;DR: In this paper, a more general class of pseudo-differential operators for non-elliptic problems is discussed. But their value is rather limited in genuinely nonelliptical problems.
Abstract: Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value is rather limited in genuinely non-elliptic problems. In this paper we shall therefore discuss some more general classes of operators which are adapted to such applications. For these operators we shall develop a calculus which is almost as smooth as that of pseudo-differential operators. It also seems that one gains some more insight into the theory of pseudo-differential operators by considering them from the point of view of the wider classes of operators to be discussed here so we shall take the opportunity to include a short exposition.
2,450 citations
TL;DR: In this paper, the authors consider l'existence and the regularite des solutions d'equations aux derivees partielles, the construction de solutions fondamentales explicites and les valeurs propres d'operateurs de Schrodinger.
Abstract: On considere l'existence et la regularite des solutions d'equations aux derivees partielles, la construction de solutions fondamentales explicites et les valeurs propres d'operateurs de Schrodinger
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TL;DR: In this article, the authors obtained the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator.
Abstract: In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is achieved by means of a complete description of the singularities of the Fourier transform of the spectral function for low frequencies.
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TL;DR: In this article, Second Order Elliptic Operators on a Compact Manifold without Boundary have been studied, as well as the Mixed Dirichlet-Cauchy Problem for Second Order Operators.
Abstract: Contents: Second Order Elliptic Operators.- Pseudo-Differential Operators.- Elliptic Operators on a Compact Manifold without Boundary.- Boundary Problems for Elliptic Differential Operators.- Symplectic Geometry.- Some Classes of (Micro-)Hypoelliptic Operators.- The Strictly Hyperbolic Cauchy Problem.- The Mixed Dirichlet-Cauchy Problem for Second Order Operators.- Appendix B: Some Spaces of Distributions.- Appendix C: Some Tools from Differential Geometry.- Bibliography.- Index.- Index of Notation.
285 citations