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

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