A parametric logarithmic improvement of the critical Hardy inequality and stability of the deficit

Ghaliah Alhamzi; Wael Mahmoud Mohammad Salameh; Prakash Jadhav; Mdi Begum Jeelani; Ghaliah Alhamzi; Wael Mahmoud Mohammad Salameh; Prakash Jadhav; Mdi Begum Jeelani

doi:10.3934/math.2026574

AIMS Mathematics

2026, Volume 11, Issue 5: 13963-13980. doi: 10.3934/math.2026574

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Research article Topical Sections

A parametric logarithmic improvement of the critical Hardy inequality and stability of the deficit

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
2.
Faculty of Information Technology, Abu Dhabi University, Abu Dhabi, United Arab Emirates
3.
Department of Mechanical Engineering, SRM University AP, Andhra Pradesh, India

Received: 20 February 2026 Revised: 08 April 2026 Accepted: 16 April 2026 Published: 18 May 2026
MSC : 26D10, 26D15, 35A23, 35P15, 47A63

We address a classification problem for critical one-dimensional Hardy forms perturbed by a logarithmic remainder. On $ (0, 1) $, with the gauge $ \log\!\frac{e}{x} = \log(e/x) $, we construct an explicit one-parameter family of Riccati weights that yields an identity-level ground-state representation. This produces a continuum of logarithmic improvements of the critical Hardy inequality with a computable remainder coefficient and an explicit positive ground state. We then derive a quantitative interior stability estimate: The Hardy deficit controls the distance to the associated ground state on every interior subinterval. We further classify the constant-coefficient logarithmic remainder class by reducing the associated ground-state ordinary differential equation (ODE) to a Euler equation in the logarithmic variable, and we obtain an interior compactness statement for sequences with vanishing deficit. As an application, we prove positivity and a priori bounds for a class of Dirichlet Schrödinger problems with critical singular potentials.
- Hardy inequality,
- logarithmic remainder,
- ground-state representation,
- Riccati equation,
- deficit stability,
- critical Schrödinger operator
Citation: Ghaliah Alhamzi, Wael Mahmoud Mohammad Salameh, Prakash Jadhav, Mdi Begum Jeelani. A parametric logarithmic improvement of the critical Hardy inequality and stability of the deficit[J]. AIMS Mathematics, 2026, 11(5): 13963-13980. doi: 10.3934/math.2026574

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Abstract

We address a classification problem for critical one-dimensional Hardy forms perturbed by a logarithmic remainder. On $ (0, 1) $, with the gauge $ \log\!\frac{e}{x} = \log(e/x) $, we construct an explicit one-parameter family of Riccati weights that yields an identity-level ground-state representation. This produces a continuum of logarithmic improvements of the critical Hardy inequality with a computable remainder coefficient and an explicit positive ground state. We then derive a quantitative interior stability estimate: The Hardy deficit controls the distance to the associated ground state on every interior subinterval. We further classify the constant-coefficient logarithmic remainder class by reducing the associated ground-state ordinary differential equation (ODE) to a Euler equation in the logarithmic variable, and we obtain an interior compactness statement for sequences with vanishing deficit. As an application, we prove positivity and a priori bounds for a class of Dirichlet Schrödinger problems with critical singular potentials.

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