The classical Hardy inequality holds in Sobolev spaces W01,p when 1 ≤ p < N. In the limiting case where p = N, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in W01,N. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as p N. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W02,p with p ∈ (1, N2 ) and the Poincaré inequality.
|Publication status||Published - 2019 Jul 22|
- Hardy inequality
- Limiting case
- Pointwise estimate of radial functions
- Sobolev embedding
ASJC Scopus subject areas