Remainder terms of $L^p$-Hardy inequalities with magnetic fields: the case $1<p<2$

TL;DR

This paper develops new remainder estimates for magnetic $L^p$-Hardy inequalities in the case $1<p<2$, introducing magnetic gradient and logarithmic terms, and addresses an open question in the field.

Contribution

It introduces novel remainder terms involving magnetic gradients and logarithmic factors for magnetic $L^p$-Hardy inequalities, extending classical results and solving an open problem.

Findings

Established new remainder terms involving magnetic gradients.

Derived logarithmic remainder terms for magnetic Hardy inequalities.

Extended results to other $L^p$-Hardy inequalities using similar methods.

Abstract

This paper focuses on remainder estimates of the magnetic $L^p$-Hardy inequalities for $1<p<2$. \emph{Firstly}, we establish a family of remainder terms involving magnetic gradients of the magnetic $L^p$-Hardy inequalities, which are also new even for the classical $L^p$-Hardy inequalities. \emph{Secondly}, we study another family of remainder terms involving logarithmic terms of the magnetic $L^p$-Hardy inequalities. \emph{Lastly}, as a byproduct, we further obtain remainder terms of some other $L^p$-Hardy-type inequalities by using similar proof of our main results. Furthermore, this paper answers the open question proposed by Cazacu \emph{et al.} in [\emph{Nonlinearity} \textbf{37} (2024), Paper No. 035004] and can be viewed as a supplementary work of it.