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  "body": "\n## Definition\n\nLet $n \\ge 1$, and let $\\Phi \\colon \\R^n \\times \\R \\to [0, \\infty)$ be the fundamental solution of the [[heat equation|VP2HN86VAG]] $\\partial_t u = \\Delta u$:\n$$\\Phi(x, t) = (4\\pi t)^{-n/2} \\exp\\!\\left(-\\frac{\\abs{x}^2}{4t}\\right) \\text{ for } t \u003e 0, \\qquad \\Phi(x, t) = 0 \\text{ for } t \\le 0.$$\nLet $\\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\\R^n \\times \\R$.\nThe *heat potential* of $\\mu$ is the function $\\Phi\\mu \\colon \\R^n \\times \\R \\to [0, \\infty]$ given by\n$$\\Phi\\mu(x, t) = \\int_{\\R^n \\times \\R} \\Phi(x - y, t - s) \\, d\\mu(y, s).$$\nWhen $d\\mu = f \\, dy \\, ds$ for a function $f \\ge 0$, the heat potential $\\Phi\\mu$ is also called the volume potential of $f$.\n\n## Remarks\n\nOnly the part of $\\mu$ with $s \u003c t$ contributes to $\\Phi\\mu(x, t)$, since $\\Phi$ vanishes for nonpositive times.\nExample: for the unit point mass $\\mu = \\delta_{(0, 0)}$, the heat potential is $\\Phi$ itself, the [[heat kernel|KD404MF042]] started at time $0$; it solves the heat equation away from $(0, 0)$.\nDegenerate case: for Lebesgue measure on $\\R^n \\times \\R$, $\\Phi\\mu(x, t) = \\int_{-\\infty}^{t} \\int_{\\R^n} \\Phi(x - y, t - s) \\, dy \\, ds = \\int_{-\\infty}^t 1 \\, ds = \\infty$ at every point, so finiteness of a heat potential needs growth conditions on $\\mu$.\nFor a bounded function $f$ on $\\R^n \\times (0, T)$ that is locally Hölder continuous, the function $u(x, t) = \\int_0^t \\int_{\\R^n} \\Phi(x - y, t - s) f(y, s) \\, dy \\, ds$ solves $\\partial_t u - \\Delta u = f$ with $u(\\cdot, 0) = 0$.\nA measure carried by the lateral boundary of a space-time cylinder gives a single layer heat potential.\nUsing a normal derivative of $\\Phi$ instead gives a double layer heat potential.\nBoth are used to solve boundary value problems for the heat equation.\n",
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