title: Hardy type inequalities in [Formula: see text] with sharp remainders.
abstract: Sharp remainder terms are explicitly given on the standard Hardy inequalities in [Formula: see text] with [Formula: see text]. Those remainder terms provide a direct and exact understanding of Hardy type inequalities in the framework of equalities as well as of the nonexistence of nontrivial extremals.
authors.affiliation: Graduate School of Science and Engineering, Ehime University, Ehime, 790-8577 Japan.
authors.forename: Norisuke
authors.initials: N
authors.lastname: Ioku
authors.affiliation: Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Osaka, 560-8531 Japan.
authors.forename: Michinori
authors.identifier: 0000-0002-5650-430X
authors.initials: M
authors.lastname: Ishiwata
authors.affiliation: Department of Applied Physics, Waseda University, Tokyo, 169-8555 Japan.
authors.forename: Tohru
authors.initials: T
authors.lastname: Ozawa
keywords: Hardy’s inequalities; remainders