Hörmander's condition

Remarks

Hörmander's theorem: under this condition, for every smooth function $c$, the operator $P = \sum_{j=1}^m X_j^2 + X_0 + c$ is hypoelliptic, where each vector field acts as a first-order differential operator. Example: on $\R^2$, let $X_1 = \partial_x$ and $X_0 = x \, \partial_y$. Then $[X_1, X_0] = \partial_y$, so $X_1$ and $[X_1, X_0]$ span $\R^2$ at every point, and the Kolmogorov operator $\partial_x^2 + x \partial_y$ is hypoelliptic. Non-example: on $\R^2$, the single field $X_1 = \partial_x$ fails the condition, since all its brackets vanish and $\partial_x$ alone spans a line. Accordingly, $\partial_x^2$ is not hypoelliptic: $u(x, y) = \abs{y}$ satisfies $\partial_x^2 u = 0$ but is not smooth. In stochastic analysis one often uses a stronger parabolic version, in which $X_0$ itself is left out of the first level of the spanning family.