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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $1 \u003c p \u003c \\infty$ and let $0 \u003c \\alpha \u003c n/p$.\nLet $\\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\\R^n$, and let $B(x, t)$ be the open ball of center $x$ and radius $t$.\nThe *Wolff potential* of $\\mu$ is the function $W^{\\mu}_{\\alpha, p} \\colon \\R^n \\to [0, \\infty]$ given by\n$$W^{\\mu}_{\\alpha, p}(x) = \\int_0^\\infty \\left( \\frac{\\mu\\big(B(x, t)\\big)}{t^{\\,n - \\alpha p}} \\right)^{1/(p - 1)} \\frac{dt}{t}.$$\nFor $R \u003e 0$, the *truncated Wolff potential* $W^{\\mu}_{\\alpha, p}(x, R)$ is the same integral over $0 \u003c t \u003c R$.\n\n## Remarks\n\nExample: let $\\mu = \\delta_0$ be the unit point mass at $0$.\nFor $x \\ne 0$, $\\mu(B(x, t)) = 0$ when $t \\le \\abs{x}$ and $= 1$ when $t \u003e \\abs{x}$, so\n$$W^{\\delta_0}_{\\alpha, p}(x) = \\int_{\\abs{x}}^\\infty t^{-\\frac{n - \\alpha p}{p - 1} - 1} \\, dt = \\frac{p - 1}{n - \\alpha p}\\, \\abs{x}^{-\\frac{n - \\alpha p}{p - 1}},$$\nwhile $W^{\\delta_0}_{\\alpha, p}(0) = \\infty$.\nFor $p = 2$, exchanging the order of integration gives $W^{\\mu}_{\\alpha, 2}(x) = \\frac{1}{n - 2\\alpha} \\int \\abs{x - y}^{2\\alpha - n} \\, d\\mu(y)$, a multiple of the [[Riesz potential|1V2N8MFSHC]] of order $2\\alpha$ of $\\mu$.\nThe condition $\\alpha \u003c n/p$ is needed: if $\\alpha p \\ge n$ and $\\mu \\ne 0$, then $\\mu(B(x, t))$ is bounded below for large $t$, the integrand does not decay fast enough, and $W^{\\mu}_{\\alpha, p} = \\infty$ everywhere.\nThe truncated potential has no such problem and is used for local estimates.\nThe case $\\alpha = 1$ is the one tied to the $p$-Laplace equation.\n",
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