@article{Alhamzi2026, 
author = {Ghaliah Alhamzi and Wael Mahmoud Mohammad Salameh and Prakash Jadhav and Mdi Begum Jeelani},
title = {A parametric logarithmic improvement of the critical Hardy inequality and stability of the deficit},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {13963-13980},
keywords = {Riccati equation, Hardy inequality, logarithmic remainder, ground-state representation, deficit stability, critical Schrödinger operator},
url = {https://www.sciopen.com/article/10.3934/math.2026574},
doi = {10.3934/math.2026574},
abstract = {We address a classification problem for critical one-dimensional Hardy forms perturbed by a logarithmic remainder. On    (  0  ,  1  ), with the gauge    log        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.}
}