---
tag: PZRK2EPH1Y
kind: definition
name: Hörmander's condition
msc: [35H10, 35H20]
defines: 91RZ5YARTM
uses: [0299VCW1TH, 68FGQEBD6J, K9DSR21DQC]
sources:
  - url: https://en.wikipedia.org/wiki/H%C3%B6rmander%27s_condition
---

## Definition

Let $\Omega \subseteq \R^n$ be an open set and let $X_0, X_1, \ldots, X_m$ be smooth [[vector fields|0299VCW1TH]] on $\Omega$.
Let $[\cdot, \cdot]$ be the [[Lie bracket of vector fields|68FGQEBD6J]], and let the indices $j_1, j_2, \ldots$ range over $\set{0, 1, \ldots, m}$.
The family satisfies *Hörmander's condition* if, at every point $x \in \Omega$, the vectors
$$X_{j_1}(x), \quad [X_{j_1}, X_{j_2}](x), \quad [X_{j_1}, [X_{j_2}, X_{j_3}]](x), \quad \ldots$$
span $\R^n$.
Equivalently, the Lie algebra generated by $X_0, \ldots, X_m$ spans the tangent space at every point.

## 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|K9DSR21DQC]], 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.