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Fractional Hardy--Maz'ya inequality on a half-space
Abstract
The main purpose of this article is to provide a fractional counterpart of the well-known Maz'ya inequality on the half-space, that is ∫ℝd+∫ℝd+(|u(x)−u(y)|p)/(|x−y|d+sp)dy dx≥𝒟d,s,p∫ℝd+|u(x)|p/xdspdx+Cd,s,p,τ∫ℝd+|u(x)|p/(xdsp−τ(xd−12+xd2)τ/2)dx, where 𝒟d,s,p stands for the sharp constant in the fractional Hardy inequality on a half-space ℝd+. We also obtain a similar result in the setting of Sobolev--Bregman forms.
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BibTeX
@misc{kijaczko2026fractional,
title = {{Fractional Hardy--Maz'ya inequality on a half-space}},
author = {Michał Kijaczko and Antoni Szczukiewicz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10832},
eprint = {2609.10832},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10832},
}
RIS
TY - UNPB
TI - Fractional Hardy--Maz'ya inequality on a half-space
AU - Michał Kijaczko
AU - Antoni Szczukiewicz
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10832
UR - https://arxiv.org/abs/2609.10832
AB - The main purpose of this article is to provide a fractional counterpart of the well-known Maz'ya inequality on the half-space, that is ∫_ℝ^d_+∫_ℝ^d_+(|u(x)−u(y)|^p)/(|x−y|^d+sp)dy dx≥𝒟_d,s,p∫_ℝ^d_+|u(x)|^p/x_d^spdx+C_d,s,p,τ∫_ℝ^d_+|u(x)|^p/(x_d^sp−τ(x_d−1^2+x_d^2)^τ/2)dx, where 𝒟_d,s,p stands for the sharp constant in the fractional Hardy inequality on a half-space ℝ^d_+. We also obtain a similar result in the setting of Sobolev--Bregman forms.
ER -