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  "body": "\n## Statement\n\nLet $n \\ge 2$, write points of $\\R^n$ as $x = (x', x_n)$ with $x' \\in \\R^{n-1}$, and let $H = \\set{x \\in \\R^n : x_n \u003e 0}$ be the upper half-space.\nLet $\\sigma_{n-1}$ be the [[surface area|0WQE3G7B88]] of the unit sphere in $\\R^n$, and for $x \\in H$ and $y \\in \\R^{n-1}$ put\n$$P(x, y) = \\frac{2 x_n}{\\sigma_{n-1} \\left( \\abs{x' - y}^2 + x_n^2 \\right)^{n/2}}.$$\nLet $g \\colon \\R^{n-1} \\to \\R$ be bounded and [[continuous|0X3WGX95FE]], and define\n$$u(x) = \\int_{\\R^{n-1}} P(x, y) \\, g(y) \\, dy, \\qquad x \\in H.$$\nThen $u$ is bounded and [[harmonic|0DEB0JMN0W]] on $H$, and $u(x) \\to g(y_0)$ as $x \\to (y_0, 0)$ with $x \\in H$, for every $y_0 \\in \\R^{n-1}$.\n\n## Remarks\n\nProof idea: for fixed $y$, $P(\\cdot, y)$ is harmonic in $H$.\nIt is, up to a constant factor, the normal derivative at the boundary of the Green's function of $H$ built by reflection.\nThe kernel is positive and $\\int_{\\R^{n-1}} P(x, y) \\, dy = 1$, so $\\abs{u} \\le \\sup \\abs{g}$.\nAs $x_n \\to 0$ the kernel concentrates at $x'$, which gives the boundary values.\nFor $n = 2$ the kernel is $\\frac{1}{\\pi} \\frac{x_2}{(x_1 - y)^2 + x_2^2}$, the Poisson kernel of the upper half-plane.\nBoundedness matters for uniqueness: $u(x) = x_n$ is harmonic on $H$ with boundary values $0$, but it is not the Poisson integral of $0$.\n\n## Explanations\n\n### L1 human\n\nThis formula fills in a harmonic function on a half-space from its values on the flat boundary.\nThe value at a point is a weighted average of the boundary values, with weights concentrated below the point.\nThe closer the point is to the boundary, the sharper the concentration.\n\n### L2 human\n\nThe half-space formula is the model for boundary behaviour of harmonic functions, nontangential limits and Hardy spaces.\nIn $\\R^2$ it is the link between harmonic functions on the upper half-plane and the Hilbert transform.\nThe family $P(\\cdot, y)$ for $x_n = t$ fixed is the Poisson semigroup, an approximate identity used throughout harmonic analysis.\n",
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