Remarks on a limiting case of hardy type inequalities

The classical Hardy inequality holds in Sobolev spaces W₀^{1, p} when 1 ≤ p < N . In the limiting case where p = N , it is known that by introducing a logarithmic weight function in the Hardy potential, some inequality which is called the critical Hardy inequality holds in W₀^1,N . In this note, in order to give an explanation of the appearance of the logarithmic function in the potential, we derive the logarithmic function from the classical Hardy inequality with best constant via some limiting procedure as p ↗ N . We show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W₀^{2, p} with p ∈ (1, ^N₂ ) and the Poincaré inequality.