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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $P$ be a [[linear differential operator|6DZETZ5T9G]] on $\\Omega$ with [[smooth|0V33BNKZ9S]] complex coefficients.\nA distribution on an open set $U$ is a continuous linear functional on the space of [[test functions|HS65DW0HXH]] on $U$.\nThe operator $P$ is *locally solvable at* $x_0 \\in \\Omega$ if there is an open neighbourhood $U \\subseteq \\Omega$ of $x_0$ with the following property.\nFor every smooth function $f$ on $\\Omega$ there is a distribution $u$ on $U$ with $P u = f$ in $U$, in the sense of distributions.\nIt is *locally solvable in* $\\Omega$ if it is locally solvable at every point of $\\Omega$.\n\n## Remarks\n\nExample: $P = \\partial_1$ on $\\R^n$ is locally solvable at every point: $u(x) = \\int_0^{x_1} f(s, x_2, \\dots, x_n) \\, ds$ is smooth and satisfies $\\partial_1 u = f$.\nMore generally, every nonzero operator with constant coefficients is locally solvable, by the [[Malgrange–Ehrenpreis theorem|3173Y813V7]].\nNon-example: Lewy's operator $L = \\partial_x + i \\partial_y - 2 i (x + i y) \\partial_t$ on $\\R^3$ with variables $(x, y, t)$.\nThere is a smooth function $f$ on $\\R^3$ such that $L u = f$ has no distribution solution in any open subset of $\\R^3$, so $L$ is locally solvable at no point.\nSome texts state the definition with slightly different classes of right-hand sides or of solutions.\n\n## Explanations\n\n### L1 human\n\nAn operator is locally solvable at a point when, for every smooth right-hand side, the equation can be solved on a fixed small neighbourhood of that point.\n",
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