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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of hypoelliptic differential operators with variable coefficients


Author: R. E. White
Journal: Proc. Amer. Math. Soc. 46 (1974), 375-382
MSC: Primary 35H05
DOI: https://doi.org/10.1090/S0002-9939-1974-0358059-0
MathSciNet review: 0358059
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Abstract: Let $ P$ be a linear differential operator with coefficients in $ {C^\infty }(\Omega )$ where $ \Omega \subset {{\mathbf{R}}^n}$. We characterize the hypoelliptic operators in terms of the $ \ast $-hypoelliptic operators. $ P$ is defined to be $ \ast $-hypoelliptic on $ \Omega $ if and only if $ u \in {\mathcal{D}'_F}(\Omega )$ and $ Pu \in {C^\infty }(\Omega )$ imply $ u \in {C^\infty }(\Omega )$. We characterize the $ \ast $-hypoelliptic operators via a priori estimates. We prove $ P$ is hypoelliptic on $ \Omega $ if and only if for $ u \in \mathcal{D}'(\Omega )$ and $ Pu \in {C^\infty }(\Omega ')$ with $ \Omega ' \subset \Omega $, there exists for each $ {x_0} \in \Omega '$ a relatively compact open neighborhood $ {\Omega _{{x_0}}} \subset \Omega '$ of $ {x_0}$ such that $ P$ is $ \ast $-hypoelliptic on $ {\Omega _{{x_0}}}$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0358059-0
Keywords: Hypoelliptic on $ \Omega $, $ \ast $-hypoelliptic on $ \Omega $, Fréchet space, open mapping theorem, local on $ \Omega $, $ \ast $-local on $ \Omega $, inductive limit
Article copyright: © Copyright 1974 American Mathematical Society