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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an open set and let $X_0, X_1, \\ldots, X_m$ be smooth [[vector fields|0299VCW1TH]] on $\\Omega$.\nLet $[\\cdot, \\cdot]$ be the [[Lie bracket of vector fields|68FGQEBD6J]], and let the indices $j_1, j_2, \\ldots$ range over $\\set{0, 1, \\ldots, m}$.\nThe family satisfies *Hörmander's condition* if, at every point $x \\in \\Omega$, the vectors\n$$X_{j_1}(x), \\quad [X_{j_1}, X_{j_2}](x), \\quad [X_{j_1}, [X_{j_2}, X_{j_3}]](x), \\quad \\ldots$$\nspan $\\R^n$.\nEquivalently, the Lie algebra generated by $X_0, \\ldots, X_m$ spans the tangent space at every point.\n\n## Remarks\n\nHörmander'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.\nExample: on $\\R^2$, let $X_1 = \\partial_x$ and $X_0 = x \\, \\partial_y$.\nThen $[X_1, X_0] = \\partial_y$, 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.\nNon-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.\nAccordingly, $\\partial_x^2$ is not hypoelliptic: $u(x, y) = \\abs{y}$ satisfies $\\partial_x^2 u = 0$ but is not smooth.\nIn 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.\n",
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