Optimal Non-Homogeneous Improvements of the Anisotropic Hardy Inequality
Abstract
We consider the Hardy inequality associated with a strongly convex Finsler norm F on ℝn and we study its optimal non-homogeneous improvements in the whole range 1<p<∞. We show that the anisotropic Hardy difference controls a weighted critical Sobolev norm when 1<p<n, a weighted exponential integrability of Trudinger type when p=n, where the Hardy term is replaced by its Leray logarithmic correction, and a weighted Hölder seminorm when p>n, proving in each regime that the logarithmic weights appear with the optimal exponents. In the borderline case p=n this extends to general functions the optimal exponent previously known only for anisotropically radial functions. The proofs rely on a common ground-state reduction, combined with a splitting into an Fo-radial and a mean-free part for 1<p≤ n, and with an anisotropic Morrey representation formula for p>n.
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BibTeX
@misc{blasio2026optimal,
title = {{Optimal Non-Homogeneous Improvements of the Anisotropic Hardy Inequality}},
author = {Giuseppina di Blasio and Giovanni Pisante and Georgios Psaradakis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33346},
eprint = {2610.33346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33346},
}
RIS
TY - UNPB
TI - Optimal Non-Homogeneous Improvements of the Anisotropic Hardy Inequality
AU - Giuseppina di Blasio
AU - Giovanni Pisante
AU - Georgios Psaradakis
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33346
UR - https://arxiv.org/abs/2610.33346
AB - We consider the Hardy inequality associated with a strongly convex Finsler norm F on ℝ^n and we study its optimal non-homogeneous improvements in the whole range 1<p<∞. We show that the anisotropic Hardy difference controls a weighted critical Sobolev norm when 1<p<n, a weighted exponential integrability of Trudinger type when p=n, where the Hardy term is replaced by its Leray logarithmic correction, and a weighted Hölder seminorm when p>n, proving in each regime that the logarithmic weights appear with the optimal exponents. In the borderline case p=n this extends to general functions the optimal exponent previously known only for anisotropically radial functions. The proofs rely on a common ground-state reduction, combined with a splitting into an F_o-radial and a mean-free part for 1<p≤ n, and with an anisotropic Morrey representation formula for p>n.
ER -