This paper focuses on remainder estimates of the magnetic \(L^p\) -Hardy inequalities for \(1<p<2\) . 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. Secondly, we study another family of remainder terms involving logarithmic terms of the magnetic \(L^p\) -Hardy inequalities. 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 et al. in [Nonlinearity 37:035004, 2024] and can be viewed as a supplementary work of it.
