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Notes on the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’

Journal of Inequalities and Applications20142014:253

https://doi.org/10.1186/1029-242X-2014-253

Received: 22 February 2014

Accepted: 17 June 2014

Published: 22 July 2014

The purpose of this note is to clarify the novelty of the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’ which was published in the J. Inequal. Appl. 2013:381, 2013 [1]. After this paper was published, the authors were informed of the references [25], and [6], the results of which partly overlap with those of [1]. In the paper [1], the authors established the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces H p , q n p ( R n ) ; see Section 2 in [1] for the precise definition of H p , q n p ( R n ) . The main theorem in [1] is stated as follows.

Theorem A [[1], Theorem 1.1]

Let n N , 1 < p < , 1 < q and 1 < α , β < . Then the inequality
( { | x | < 1 2 } | u ( x ) | α | log | x | | β d x | x | n ) 1 α C u H p , q n p
(1)
holds for all u H p , q n p ( R n ) if and only if one of the following conditions (i), (ii), and (iii) is fulfilled:
{ i i ( i ) 1 + α β < 0 ; i ( ii ) 1 + α β 0 and q < α 1 + α β ; ( iii ) 1 + α β > 0 , q = α 1 + α β and α β .
(2)

However, the inequality (1) was established under the condition (iii) in [7] and [6]. More precisely, the particular case of α = β = p = q was considered in [7], where the authors also obtained similar inequalities in the critical Besov spaces. In [6], the authors considered Bessel potential spaces with a logarithmic smoothness on Lorentz-Zygmund spaces which extend the critical Sobolev-Lorentz spaces H p , q n p ( R n ) for instance. Indeed, the authors in [6] proved logarithmic Hardy inequalities corresponding to (1) in critical Sobolev-Lorentz-Zygmund spaces with a logarithmic smoothness in [[6], Theorem 5.1]. This theorem includes the inequality (1) under the condition (iii) as special cases.

Based on the historical remarks above, the novelty in the paper [1] is to have given other conditions (i) and (ii) with which the inequality (1) holds and to have shown that the conditions (i), (ii), and (iii) are also necessary for (1). Especially, the marginal case q = was also considered in [1], where the norm u H p , n p becomes smallest in the sense of Sobolev-Lorentz spaces, and Theorem A shows that the condition (i) is necessary and sufficient for (1) to hold.

We also add other references [24], and [5] in this note, which are closely related to the results in [1] and which were not cited in [1].

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Saitama University
(2)
Department of Applied Physics, Waseda University
(3)
Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University

References

  1. Machihara S, Ozawa T, Wadade H: Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces. J. Inequal. Appl. 2013. Article ID 381, 2013: Article ID 381Google Scholar
  2. Gogatishvili A, Neves JS, Opic B: Optimality of embeddings of Bessel-potential-type spaces into Lorentz-Karamata spaces. Proc. R. Soc. Edinb., Sect. A, Math. 2004, 134: 1127-1147. 10.1017/S0308210500003668MathSciNetView ArticleMATHGoogle Scholar
  3. Gogatishvili A, Neves JS, Opic B: Optimality of embeddings of Bessel-potential-type spaces. In FSDONA 2004 Proceedings. Edited by: Drábek P, Rákosník J. Math. Inst. Acad. Sci. Czech Republic, Prague; 2005:97-112.Google Scholar
  4. Neves JS: Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings. Diss. Math. 2002, 405: 1-46.MathSciNetMATHGoogle Scholar
  5. Opic B, Trebels W: Bessel potentials with logarithmic components and Sobolev-type embeddings. Anal. Math. 2000, 26: 299-319. 10.1023/A:1005691512014MathSciNetView ArticleMATHGoogle Scholar
  6. Opic B, Trebels W: Sharp embeddings of Bessel potential spaces with logarithmic smoothness. Math. Proc. Camb. Philos. Soc. 2003, 134: 347-384. 10.1017/S0305004102006321MathSciNetView ArticleMATHGoogle Scholar
  7. Edmunds DE, Triebel H: Sharp Sobolev embeddings and related Hardy inequalities: the critical case. Math. Nachr. 1999, 207: 79-92.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Machihara et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.