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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^n$ be a bounded open set with smooth boundary $\\Gamma = \\partial\\Omega$ and outward unit normal $\\nu$.\nLet $P$ be an [[elliptic|B4TTNHSTRX]] linear differential operator of order $2m$ with smooth coefficients on a neighbourhood $U$ of $\\overline{\\Omega}$.\nAssume $P$ has a [[fundamental solution|K2NK9A8M5F]] $E$ on $U$, that is, $P E = \\delta$ in the sense of distributions.\nFor $u$ smooth up to $\\Gamma$, let $\\gamma u = (u|_\\Gamma, \\partial_\\nu u|_\\Gamma, \\ldots, \\partial_\\nu^{2m-1} u|_\\Gamma)$ be its Cauchy data, given by the [[trace operator|Z2T5R88QDB]].\nGreen's formula for $P$ gives a potential operator $K$, built from $E$, such that $u = K(\\gamma u)$ in $\\Omega$ whenever $Pu = 0$ in $\\Omega$.\nLet $\\gamma_+$ take the Cauchy data of a function on $\\Omega$ as limits from inside.\nThe *Calderón projector* of $P$ and $\\Omega$ is the operator $C = \\gamma_+ K$ on $2m$-tuples of functions on $\\Gamma$.\n\n## Remarks\n\n$C$ is a $2m \\times 2m$ matrix of [[pseudo-differential operators|FF27N0BWTP]] on $\\Gamma$, $C^2 = C$, and its range is the space of Cauchy data $\\gamma u$ of the solutions of $Pu = 0$ in $\\Omega$.\nIf $E$ is only a parametrix, then $C$ is a projection up to smoothing operators, which is enough for most uses.\nExample: for $P = -\\Delta$ ($m = 1$), $C$ acts on pairs $(f, g)$.\nIts four entries are, up to signs and the factor $\\tfrac12 I$, the double layer, single layer, hypersingular and adjoint double layer operators.\nA pair $(f, g)$ is fixed by $C$ exactly when $f = u|_\\Gamma$ and $g = \\partial_\\nu u|_\\Gamma$ for a [[harmonic function|0DEB0JMN0W]] $u$ on $\\Omega$.\nNon-example: the identity on pairs $(f, g)$ is a projection, but its range is too large.\nOn the unit disk it contains $(0, 1)$, which is not Cauchy data of a harmonic $u$, since $u = 0$ on the circle forces $u = 0$.\n",
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