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Note on Hypoellipticity of a First Order Linear Partial Differential Operator

Published online by Cambridge University Press:  22 January 2016

Yoshio Kato
Yoshio Kato*
Affiliation:
Department of Mathematics, Aichi University of Education, Okazaki-shi, Aichi-ken (Japan)
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Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 323 - 330
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Hörmander, L.: Linear partial differential operators, Springer-Ver., Berlin (1963).Google Scholar
[2] Mizohata, S.: Solutions nulles et solutions non analytiques, J. Math. Kyoto Univ., 1 (1962), 271–302.Google Scholar
[3] Nirenberg, L. and Tréves, F.: Solvability of a first order linear partial differential equation, Comm. Pure Appl. Math., 16 (1963), 331–351.CrossRefGoogle Scholar
[4] Suzuki, H.: Analytic-hypoelliptic differential operators of first order in two independent variables, J. Math. Soc. Japan, 16 (1964), 367–374.CrossRefGoogle Scholar
[5] Yoshikawa, A.: On the hypoellipticity of differential operators, J. Fac. Sci. Univ. Tokyo, 14 (1967), 81–88.Google Scholar