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

Shuji Machihara; Tohru Ozawa; Hidemitsu Wadade

doi:10.1186/1029-242X-2013-381

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

Shuji Machihara, Tohru Ozawa, Hidemitsu Wadade^*

^*Corresponding author for this work

School of Advanced Science and Engineering

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

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 language	English
Article number	381
Journal	Journal of Inequalities and Applications
Volume	2013
DOIs	https://doi.org/10.1186/1029-242X-2013-381
Publication status	Published - 2013 Dec

Keywords

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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N2 - 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.

AB - 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.

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KW - Logarithmic Hardy inequality

KW - O'Neil's inequality

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

Abstract

Keywords

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