Near-Critical Asymptotics and Radial Sharpness for a Two-Weight Hardy Inequality
Abstract
In this paper, we study the optimal constant C(p,γ) in a planar two-weight Hardy inequality relating the gradient and Hessian of a scalar potential, posed by Maz'ya in [14, Problem 3.11]. For every 1≤ p<∞, we prove limγ→γc 2p/2|γ−γc|p C(p,γ)=1, γc=1−2/p. The proof reduces the curl-free constraint to an angular minimization problem and yields qualitative rigidity of almost extremizers for p>1. Moreover, we determine the sharp constant in the radial class for every 1≤ p<∞ and γ≠γc, thereby resolving the radial case of Maz'ya's problem. We also show that the radial inequality remains valid at γ=γc, with the same sharp constant Crad(p,γ) =[1+(γ−1+2/p)2]−p/2. For general, not necessarily radial, potentials, we identify explicit parameter regions where this constant remains sharp and establish two sufficient criteria for symmetry breaking. The exact transition remains open for p≠2. In the quadratic case, we recover the known sharp constant and derive a decomposition of its deficit. At the critical exponent, we obtain an exact remainder identity for the logarithmic Hardy inequality, whose classical constant remains sharp among radial potentials. The logarithmic inequality fails at p=1.
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BibTeX
@misc{shaimerdenov2026nearcritical,
title = {{Near-Critical Asymptotics and Radial Sharpness for a Two-Weight Hardy Inequality}},
author = {Yerkin Shaimerdenov and Nurgissa Yessirkegenov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02328},
eprint = {2611.02328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02328},
}
RIS
TY - UNPB
TI - Near-Critical Asymptotics and Radial Sharpness for a Two-Weight Hardy Inequality
AU - Yerkin Shaimerdenov
AU - Nurgissa Yessirkegenov
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02328
UR - https://arxiv.org/abs/2611.02328
AB - In this paper, we study the optimal constant C(p,γ) in a planar two-weight Hardy inequality relating the gradient and Hessian of a scalar potential, posed by Maz'ya in [14, Problem 3.11]. For every 1≤ p<∞, we prove lim_γ→γ_c 2^p/2|γ−γ_c|^p C(p,γ)=1, γ_c=1−2/p. The proof reduces the curl-free constraint to an angular minimization problem and yields qualitative rigidity of almost extremizers for p>1. Moreover, we determine the sharp constant in the radial class for every 1≤ p<∞ and γ≠γ_c, thereby resolving the radial case of Maz'ya's problem. We also show that the radial inequality remains valid at γ=γ_c, with the same sharp constant C_rad(p,γ) =[1+(γ−1+2/p)^2]^−p/2. For general, not necessarily radial, potentials, we identify explicit parameter regions where this constant remains sharp and establish two sufficient criteria for symmetry breaking. The exact transition remains open for p≠2. In the quadratic case, we recover the known sharp constant and derive a decomposition of its deficit. At the critical exponent, we obtain an exact remainder identity for the logarithmic Hardy inequality, whose classical constant remains sharp among radial potentials. The logarithmic inequality fails at p=1.
ER -