Preprint

Inserted: 18 aug 2026
Last Updated: 18 aug 2026

Year: 2026
Links: PDF on ResearchGate

Abstract:

We study the far-field decay of capacitary potentials associated with measurable, symmetric kernels $K$ of fractional $p$-Laplacian type, for $p\in(1,\infty)$. If $n\geq 1$ denotes the space-dimension and $s\in (0,1)$ the kernel fractional order, in the regime $sp<n$, we prove that the capacitary potential $U$ of any compact subset of $\mathbb{R}^n$ vanishes at infinity, and we construct a finite capacitary measure associated with it. For $n\ge2$, our main result is a two-sided global Wolff-potential estimate of $U$ in terms of the Wolff potential of its capacitary measure. As a consequence, we get the sharp polynomial far-field decay \[ U(x) \asymp {\text{Cap}}_{K,s,p}(F)^{\frac{1}{p-1}}
x
^{-\frac{n-sp}{p-1}} \ \text{as}\
x
\to+\infty, \] for every compact set $F\subset \mathbb{R}^n$, where ${\text{Cap}}_{K,s,p}$ denotes the fractional $K$-capacity.

In the case $p=2$ and for the Gagliardo kernel, we additionally obtain an exact Riesz representation through an equilibrium measure. We identify the total mass of such a measure, including the normalization constant of the fractional Laplacian, and we derive the exact leading asymptotic profile at infinity.

On account of the results above, we also prove a finite-energy analogue of Serrin's removable-singularity criterion: a compact set $F$ has zero fractional $K$-capacity if and only if every finite-energy $K$-harmonic function in $\mathbb R^n\setminus F$ is $K$-harmonic in $\mathbb R^n$.

Keywords: fractional Laplacian, fractional capacity, capacitary potentials, Wolff potentials, Riesz representation, removability, far-field asymptotics