functional analysis

Sharp forms and quantitative stability for general weighted discrete -Hardy inequalities

arXiv:2604.02229

summary

The paper derives a sharp remainder term for general weighted discrete p‑Hardy inequalities and shows that the inequality’s deficit quantitatively controls the distance to the set of minimizers.

Abstract

In this paper, we provide a sharp remainder term for the general weighted discrete -Hardy inequality. By choosing appropriate weights and specifying , we are able to recover the identity by Krej{č}i{ř}{\'ı}k-Štampach [KS22, Theorem 1], obtain the sharp form of the -Hardy inequality by Fischer-Keller-Pogorzelski [FKP23, Theorem 1] and generalize the power weighted inequality by Gupta [Gup22, Theorem 2.1] with a sharp remainder. In addition, we prove a quantitative stability type result, thereby showing that the deficit of the discrete -Hardy inequality controls the weighted distance to the family of non-trivial minimizers.

24 pages, revised version with additional details, updated references, and improvements to the exposition

Topics & keywords

#hardy inequalities#discrete inequalities#weighted inequalities#quantitative stability#sharp constantsp‑Hardy inequalitysharp remainder termweighted discrete inequalitydeficit estimateminimizer stability
Sharp forms and quantitative stability for general weighted discrete $p$-Hardy inequalities · wovepaper