# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Solutions of partial differential equations with support on leaves of associated foliationsHTML articles powered by AMS MathViewer

by E. C. Zachmanoglou
Trans. Amer. Math. Soc. 180 (1973), 415-421 Request permission

## Abstract:

Suppose that the linear partial differential operator \$P(x,D)\$ has analytic coefficients and that it can be written in the form \$P(x,D) = R(x,D)S(x,D)\$ where \$S(x,D)\$ is a polynomial in the homogeneous first order operators \${A_1}(x,D), \cdots ,{A_r}(x,D)\$. Then in a neighborhood of any point \${x^0}\$ at which the principal part of \$S(x,D)\$ does not vanish identically, there is a solution of \$P(x,D)u = 0\$ with support the leaf through \${x^0}\$ of the foliation induced by the Lie algebra generated by \${A_1}(x,D), \cdots ,{A_r}(x,D)\$. This result yields necessary conditions for hypoellipticity and uniqueness in the Cauchy problem. An application to second order degenerate elliptic operators is also given.
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