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Transactions of the American Mathematical Society

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



Gelfand theory of pseudo differential operators and hypoelliptic operators

Author: Michael E. Taylor
Journal: Trans. Amer. Math. Soc. 153 (1971), 495-510
MSC: Primary 47G05; Secondary 35H05
MathSciNet review: 0415430
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Abstract: This paper investigates an algebra $ \mathfrak{A}$ of pseudo differential operators generated by functions $ a(x) \in {C^\infty }({R^n}) \cap {L^\infty }({R^n})$ such that $ {D^\alpha }a(x) \to 0$ as $ \vert x\vert \to \infty $, if $ \vert\alpha \vert \geqq 1$, and by operators $ q(D){Q^{ - 1/2}}$ where $ q(D) < P(D),Q = I + P{(D)^ \ast }P(D)$, and $ P(D)$ is hypoelliptic. It is proved that such an algebra has compact commutants, and the maximal ideal space of the commutative $ {C^ \ast }$ algebra $ \mathfrak{A}/J$ is investigated, where $ J$ consists of the elements of $ \mathfrak{A}$ which are compact. This gives a necessary and sufficient condition for a differential operator $ q(x,D):{\mathfrak{B}_2}_{,\tilde P} \to {L^2}$ to be Fredholm. (Here and in the rest of this paragraph we assume that the coefficients of all operators under consideration satisfy the conditions given on $ a(x)$ in the first sentence.) It is also proved that if $ p(x,D)$ is a formally selfadjoint operator on $ {R^n}$ which has the same strength as $ P(D)$ uniformly on $ {R^n}$, then $ p(x,D)$ is selfadjoint, with domain $ {\mathfrak{B}_{2,\tilde P}}({R^n})$, and semibounded, if $ n \geqq 2$. From this a Gårding type inequality for uniformly strongly formally hypoelliptic operators and a global regularity theorem for uniformly formally hypoelliptic operators are derived. The familiar local regularity theorem is also rederived.

It is also proved that a hypoelliptic operator $ p(x,D)$ of constant strength is formally hypoelliptic, in the sense that for any $ {x_0}$, the constant coefficients operator $ p({x_0},D)$ is hypoelliptic.

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Keywords: Pseudo differential operator, hypoelliptic operator, Fredholm operator
Article copyright: © Copyright 1971 American Mathematical Society