Let P be a partial differential operator defined in an open subset Un. If for every distributionPlanetmathPlanetmath (http://planetmath.org/Distribution4) u defined in an open subset VU such that Pu is C (smooth), u must also be C, then P is called hypoelliptic.

Similarly, if the same assertion holds with C replaced by real analytic, then P is said to be analytically hypoelliptic.

Note that some authors use “hypoelliptic” to mean “analytically hypoelliptic.” Hence, if it is not clear from context, it is best to specify the regularity when using the term. For example, C-hypoelliptic instead of just hypoelliptic.


  • 1 J. Barros-Neto, Ralph A. Artino. , Lecture Notes in Pure and Applied Mathematics, 53. Marcel Dekker, Inc., New York, 1980. http://www.ams.org/mathscinet-getitem?mr=81k:35031MR 81k:35031
  • 2 Bernard Helffer, Francis Nier. , Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005. http://www.ams.org/mathscinet-getitem?mr=2006a:58039MR 2006a:58039
  • 3 Norio Shimakura. , Kinokuniya Company Ltd., Tokyo, Japan, 1978.
Title hypoelliptic
Canonical name Hypoelliptic
Date of creation 2013-03-22 16:01:13
Last modified on 2013-03-22 16:01:13
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 35H10
Defines analytically hypoelliptic
Defines analytic hypoelliptic