Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces

Shuji Machihara, Tohru Ozawa, Hidemitsu Wadade*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


In this paper, we establish the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces. More precisely, we generalize the Hardy type inequality obtained in Edmunds and Triebel (Math. Nachr. 207:79-92, 1999). The generalized inequality allows us to take the exponents appearing in the inequality more flexibly, and its optimality is discussed in detail. O'Neil's inequality and its reverse play an essential role for the proof.

Original languageEnglish
Article number381
JournalJournal of Inequalities and Applications
Publication statusPublished - 2013 Dec


  • Critical Sobolev-Lorentz space
  • Logarithmic Hardy inequality
  • O'Neil's inequality

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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