Sharp remainder of a critical Hardy inequality

By Norisuke Ioku, M. Ishiwata, Tohru Ozawa. 2016

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On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups (2016, background): “, [19] and [32] for discussions in Euclidean cases).”
Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups (2016, methodology): “ties (1.1) and (1.4) (and probably also (1.2)) fail for any constant. The critical version of (1.1) with p = n was investigated by Edmunds and Triebel [ET99], and of (1.3) by Ioku, Ishiwata and Ozawa [IIO16]. Their generalisations as well as a number of other critical (logarithmic) Hardy inequalitie”
Scale invariance structures of the critical and the subcritical Hardy inequalities and their improvements (2017, background): “We remark again that another form of remainder term is obtained, see [17,20].”