Hypoelliptic operator

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title: "Hypoelliptic operator" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["partial-differential-equations", "differential-operators"] description: "Partial differential operator" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Hypoelliptic_operator" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Partial differential operator ::

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

:U \subset{\mathbb{R}}^n

is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smooth), u must also be C^\infty.

If this assertion holds with C^\infty replaced by real-analytic, then P is said to be analytically hypoelliptic.

Every elliptic operator with C^\infty coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). In addition, the operator for the heat equation (P(u)=u_t - k,\Delta u,) :P= \partial_t - k,\Delta_x, (where k0) is hypoelliptic but not elliptic. However, the operator for the wave equation (P(u)=u_{tt} - c^2,\Delta u,) : P= \partial^2_t - c^2,\Delta_x, (where c\ne 0) is not hypoelliptic.

References

• {{cite book | last = Shimakura | first = Norio | title = Partial differential operators of elliptic type: translated by Norio Shimakura | publisher = American Mathematical Society, Providence, R.I | date = 1992 | pages = | isbn = 0-8218-4556-X
• {{cite book | last = Egorov | first = Yu. V. |author2=Schulze, Bert-Wolfgang | title = Pseudo-differential operators, singularities, applications | publisher = Birkhäuser | date = 1997 | pages = | isbn = 3-7643-5484-4
• {{cite book | last = Vladimirov | first = V. S. | title = Methods of the theory of generalized functions | publisher = Taylor & Francis | date = 2002 | pages = | isbn = 0-415-27356-0
• {{cite book | last = Folland | first = G. B. | title = Fourier Analysis and its applications | publisher = AMS | date = 2009 | pages = | isbn = 978-0-8218-4790-9

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