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.