Hypoelliptic operator

In the theory of partial differential equations, a partial differential operator defined on an open subset

is called hypoelliptic if for every distribution defined on an open subset such that is (smooth), must also be .

If this assertion holds with replaced by real analytic, then is said to be analytically hypoelliptic.

Every elliptic operator with coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). The heat equation operator

(where ) is hypoelliptic but not elliptic. The wave equation operator

(where ) is not hypoelliptic.


  • Shimakura, Norio (1992). Partial differential operators of elliptic type: translated by Norio Shimakura. American Mathematical Society, Providence, R.I. ISBN 0-8218-4556-X.
  • Egorov, Yu. V.; Schulze, Bert-Wolfgang (1997). Pseudo-differential operators, singularities, applications. Birkhäuser. ISBN 3-7643-5484-4.
  • Vladimirov, V. S. (2002). Methods of the theory of generalized functions. Taylor & Francis. ISBN 0-415-27356-0.
  • Folland, G. B. (2009). Fourier Analysis and its applications. AMS. ISBN 0-8218-4790-2.

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