hypoelliptic

Definition.

Let $P$ be a partial differential operator defined in an open subset $U\subset {ℝ}^n$. If for every distribution (http://planetmath.org/Distribution4) $u$ defined in an open subset $V\subset U$ such that $Pu$ is $C^{\mathrm{\infty }}$ (smooth), $u$ must also be $C^{\mathrm{\infty }}$, then $P$ is called hypoelliptic.

Similarly, if the same assertion holds with $C^{\mathrm{\infty }}$ 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, $C^{\mathrm{\infty }}$-hypoelliptic instead of just hypoelliptic.

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