Wolff potential

Remarks

Example: let $\mu = \delta_0$ be the unit point mass at $0$. For $x \ne 0$, $\mu(B(x, t)) = 0$ when $t \le \abs{x}$ and $= 1$ when $t > \abs{x}$, so $$W^{\delta_0}_{\alpha, p}(x) = \int_{\abs{x}}^\infty t^{-\frac{n - \alpha p}{p - 1} - 1} \, dt = \frac{p - 1}{n - \alpha p}\, \abs{x}^{-\frac{n - \alpha p}{p - 1}},$$ while $W^{\delta_0}_{\alpha, p}(0) = \infty$. For $p = 2$, 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 of order $2\alpha$ of $\mu$. The condition $\alpha < n/p$ 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. The truncated potential has no such problem and is used for local estimates. The case $\alpha = 1$ is the one tied to the $p$-Laplace equation.