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Mat. Zametki, 1972, Volume 12, Issue 3, Pages 269–274 (Mi mz9878)  

An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity

V. S. Fedii

Novocherkassk Polytechnic Institute

Abstract: It is proved that the operator
$$ P\equiv-\frac{\partial^2}{\partial x_1^2}-\sum_{k=2}^n\frac{\partial}{\partial x_k}\varphi^2(x)\frac\partial{\partial x_k}, $$
where $\varphi(x)\in C^\infty(\Omega)$ ($\Omega$ is a domain in $\mathbf{R}^n$), $\{x: \varphi(x)=0\}$ is a compactum in $\Omega$ which is the closure of its internal points, has the property of global hypoellipticity in $\Omega$, i.e.,
$$ v\in D'(\Omega),\qquad Pv\in C^\infty(\Omega)\Longrightarrow v\in C^\infty(\Omega). $$
This operator is not hypoelliptic.

Full text: PDF file (608 kB)

English version:
Mathematical Notes, 1972, 12:3, 595–598

Bibliographic databases:

UDC: 517.9
Received: 28.09.1971

Citation: V. S. Fedii, “An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity”, Mat. Zametki, 12:3 (1972), 269–274; Math. Notes, 12:3 (1972), 595–598

Citation in format AMSBIB
\Bibitem{Fed72}
\by V.~S.~Fedii
\paper An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity
\jour Mat. Zametki
\yr 1972
\vol 12
\issue 3
\pages 269--274
\mathnet{http://mi.mathnet.ru/mz9878}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=320501}
\zmath{https://zbmath.org/?q=an:0243.35021}
\transl
\jour Math. Notes
\yr 1972
\vol 12
\issue 3
\pages 595--598
\crossref{https://doi.org/10.1007/BF01093992}


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