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.
- Critical Sobolev-Lorentz space
- Logarithmic Hardy inequality
- O'Neil's inequality
ASJC Scopus subject areas
- Applied Mathematics
- Discrete Mathematics and Combinatorics