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A note on p-harmonic measure
Abstract
Let 1 < p < ∞. We identify the optimal decay exponent of the p-harmonic measure $ω_p ( B( ξ, t) \cap \partial \mathbb{B}, 0, \mathbb{B}) $ for small t, where 𝔹 is the unit ball in ℝn, ξ∈ ∂ 𝔹 and $B(ξ, t)$ denotes the euclidean ball centered at ξ of radius t. As a consequence and, under suitable geometrical restrictions on a domain $Ω\subset \mathbb{R}^n $, we identify the optimal decay exponent of ωp ( B( ξ, t) ∩ ∂ Ω, a, Ω) for small t, where a ∈ Ω and ξ∈ ∂ Ω. When restricted to the range 1 < p < n we provide new estimates complementing previous results of different authors in the unit ball, the unit disc or the upper half-space.
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BibTeX
@misc{llorente2026note,
title = {{A note on p-harmonic measure}},
author = {José González Llorente},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12176},
eprint = {2609.12176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12176},
}
RIS
TY - UNPB
TI - A note on p-harmonic measure
AU - José González Llorente
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.12176
UR - https://arxiv.org/abs/2609.12176
AB - Let 1 < p < ∞. We identify the optimal decay exponent of the p-harmonic measure $ω_p ( B( ξ, t) \cap \partial \mathbb{B}, 0, \mathbb{B}) $ for small t, where 𝔹 is the unit ball in ℝ^n, ξ∈ ∂ 𝔹 and $B(ξ, t)$ denotes the euclidean ball centered at ξ of radius t. As a consequence and, under suitable geometrical restrictions on a domain $Ω\subset \mathbb{R}^n $, we identify the optimal decay exponent of ω_p ( B( ξ, t) ∩ ∂ Ω, a, Ω) for small t, where a ∈ Ω and ξ∈ ∂ Ω. When restricted to the range 1 < p < n we provide new estimates complementing previous results of different authors in the unit ball, the unit disc or the upper half-space.
ER -