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  "body": "\n## Counterexample\n\nOn $\\R^3$ with coordinates $(x, y, t)$, consider the first-order linear operator\n$$L = \\frac{\\partial}{\\partial x} + i \\frac{\\partial}{\\partial y} - 2i(x + iy) \\frac{\\partial}{\\partial t}.$$\nIts coefficients are polynomials, so they are smooth and even real-analytic.\nHans Lewy proved that there is a function $f \\in C^\\infty(\\R^3)$ with the following property.\nThe [[partial differential equation|GC5BDZ6QBC]] $Lu = f$ has no solution $u$ of class $C^1$ on any nonempty open subset of $\\R^3$.\nSo a linear partial differential equation with smooth coefficients and a smooth right-hand side need not be locally solvable.\n\n## Remarks\n\nThe key step concerns right-hand sides that depend only on $t$.\nIf $\\psi$ is a real-valued $C^1$ function of $t$ and $Lu = \\psi'(t)$ has a $C^1$ solution near the origin, then $\\psi$ is real-analytic near $t = 0$.\nChoosing $\\psi$ smooth but not real-analytic gives an equation with no solution near the origin; a Baire category argument then produces one $f$ that works at every point.\nThe operator $L$ is, up to a constant factor, the [[tangential Cauchy–Riemann operator|X7GP9BECD8]] of the hypersurface $\\set{(z, w) \\in \\C^2 : \\operatorname{Im} w = \\abs{z}^2}$, written in the coordinates $z = x + iy$ and $t = \\operatorname{Re} w$.\nThis contrasts with the Cauchy–Kovalevskaya theorem, which gives local solutions when the coefficients and the data are all real-analytic.\nHörmander later explained the phenomenon by a necessary condition for local solvability involving the commutator of $L$ and its complex conjugate $\\bar L$.\n\n## Explanations\n\n### L1 human\n\nBefore this example, it was widely expected that a linear differential equation with nice coefficients always has solutions at least on small regions.\nLewy wrote down a simple equation in three real variables, with polynomial coefficients, that has no solution at all near any point for a suitable smooth right-hand side.\nThe obstruction comes from complex analysis: solutions would force certain smooth functions to be real-analytic, which they need not be.\n",
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