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  "body": "\n## Statement\n\nLet $P = \\sum_{\\abs{\\alpha} \\le m} c_\\alpha \\partial^\\alpha$ be a [[linear differential operator|6DZETZ5T9G]] on $\\R^n$ with constant complex coefficients $c_\\alpha$, not all zero.\nThen $P$ has a [[fundamental solution|K2NK9A8M5F]]: there is a distribution $E$ on $\\R^n$ with\n$$P E = \\delta,$$\nwhere $\\delta$ is the Dirac distribution at the origin.\nConsequently, for every [[test function|HS65DW0HXH]] $f$ on $\\R^n$, the function $u = E * f$ is smooth and satisfies $P u = f$.\n\n## Remarks\n\nFormally, the Fourier transform turns $P E = \\delta$ into multiplication of $\\widehat{E}$ by a polynomial $p(\\xi)$, so one wants $\\widehat{E} = 1 / p$.\nThe difficulty is that $p$ may vanish on the real space.\nOne proof avoids its zeros by moving the Fourier integration into the complex domain.\nAnother uses the Hahn–Banach theorem together with an estimate $\\norm{\\varphi} \\le C \\norm{P^{*} \\varphi}$ in suitable norms.\nThe theorem was proved independently by Bernard Malgrange and Leon Ehrenpreis.\nExample: for $P = \\Delta$ on $\\R^3$, the function $E(x) = -1 / (4 \\pi \\abs{x})$ is a fundamental solution, and $u = E * f$ solves the Poisson equation $\\Delta u = f$.\nThe hypothesis of constant coefficients is needed: Lewy's operator, which has polynomial coefficients, has no [[local solvability|P6E9RZNRBZ]] at all.\nConstant-coefficient operators are even solvable in the large, by a further classical theorem.\nThat is, $P u = f$ has a smooth solution on $\\R^n$ for every smooth $f$ on $\\R^n$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that every linear differential equation with constant coefficients can be solved for a point source.\nOnce the response $E$ to a point source is known, the response to any source $f$ is obtained by adding up shifted copies of $E$, that is, by convolution.\nSo every such equation can be solved, at least for sources that vanish outside a bounded set.\n\n### L2 human\n\nThe theorem is the starting point of the general theory of constant-coefficient operators.\nFor particular operators one writes the fundamental solution down explicitly, such as the Newtonian potential for the Laplacian or the [[heat kernel|KD404MF042]] for the heat operator.\nThe general theorem guarantees existence but gives little information on $E$; finer questions, such as whether $P$ is [[hypoelliptic|K9DSR21DQC]], depend on properties of the fundamental solution.\nThe theorem fails for variable coefficients, and the study of when local solvability holds then becomes a hard problem.\n",
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