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 Mat. Sb. (N.S.), 1971, Volume 85(127), Number 1(5), Pages 18–48 (Mi msb3180)

On a criterion for hypoellipticity

V. S. Fedii

Abstract: In this paper a criterion for hypoellipticity is proved which is formulated in terms of certain estimates in the $H_{(s)}$ norms, and which is a generalization of a criterion of Trèves. With the use of this criterion it is possible to prove the hypoellipticity of certain operators that do not satisfy Hörmander's criterion. It is proved, for example, that the operator $P=\partial^2/\partial x^2+\varphi^2(x)\partial^2/\partial y^2$ is hypoelliptic, where $\varphi(x)$ is an infinitely differentiable function that is not equal to zero for $x\ne0$ and has a zero of infinite order at $x=0$.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 14:1, 15–45

Bibliographic databases:

UDC: 517.43
MSC: 47F05

Citation: V. S. Fedii, “On a criterion for hypoellipticity”, Mat. Sb. (N.S.), 85(127):1(5) (1971), 18–48; Math. USSR-Sb., 14:1 (1971), 15–45

Citation in format AMSBIB
\Bibitem{Fed71} \by V.~S.~Fedii \paper On a~criterion for hypoellipticity \jour Mat. Sb. (N.S.) \yr 1971 \vol 85(127) \issue 1(5) \pages 18--48 \mathnet{http://mi.mathnet.ru/msb3180} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=287160} \zmath{https://zbmath.org/?q=an:0247.35023} \transl \jour Math. USSR-Sb. \yr 1971 \vol 14 \issue 1 \pages 15--45 \crossref{https://doi.org/10.1070/SM1971v014n01ABEH002602} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Grushin, “Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols”, Math. USSR-Sb., 17:4 (1972), 497–514
2. Kazuo Amano, “Hypoellipticity of a class of degenerate elliptic–parabolic operators”, Communications in Partial Differential Equations, 6:8 (1981), 903
3. Osamu Kobayashi, Akira Yoshioka, Yoshiaki Maeda, Hideki Omori, “The theory of infinite-dimensional Lie groups and its applications”, Acta Appl Math, 3:1 (1985), 71
4. Toshihiko Hoshiro, “On Levi-type conditions for hypoellipticity of certain diffrential operators”, Communications in Partial Differential Equations, 17:5-6 (1992), 905
5. G. A. Smolkin, “On a property of solutions of a special class of degenerate elliptic equations”, Russian Math. Surveys, 51:3 (1996), 561–562
6. Nguyen Minh Tri, “On Grushin's equation”, Math. Notes, 63:1 (1998), 84–93
7. J.J Kohn, “Hypoellipticity of Some Degenerate Subelliptic Operators”, Journal of Functional Analysis, 159:1 (1998), 203
8. Nguyen Minh Tri, “Some Examples of Nonhypoelliptic Infinitely Degenerate Elliptic Differential Operators”, Math. Notes, 71:4 (2002), 517–529
9. Eric T. Sawyer, Richard L. Wheeden, “ A priori estimates for quasilinear equations related to the Monge-Ampère equation in two dimensions”, J Anal Math, 97:1 (2005), 257
10. Luca Baracco, Tran Vu Khanh, Giuseppe Zampieri, “Propagation of holomorphic extendibility and non-hypoellipticity of the -Neumann problem in an exponentially degenerate boundary”, Advances in Mathematics, 230:4-6 (2012), 1972
11. Cristian Rios, E.T.. Sawyer, R.L.. Wheeden, “Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem”, JAMA, 119:1 (2013), 1
12. Lyudmila Korobenko, Cristian Rios, “Hypoellipticity of certain infinitely degenerate second order operators”, Journal of Mathematical Analysis and Applications, 2013
13. Luca Baracco, “A multiplier condition for hypoellipticity with optimal loss of derivatives of complex vector fields”, Journal of Mathematical Analysis and Applications, 2014
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