Remarks on a limiting case of Hardy type inequalities

Abstract

The classical Hardy inequality holds in Sobolev spaces W 1,p 0 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 W 1,N 0 . 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 ր N. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W 2,p 0 with p ∈ (1, N 2 ) and the Poincaré inequality.

Two limits on Hardy and Sobolev inequalities

“…Hereinafter we set f (t) = e −t . Proof of Theomre 3F From Lemma 2 [29], it is enough to show the inequality (2.4) for radial function u ∈ C 1 c (B 1 ). By the Hardy inequality (1.1), for k ≥ 1 we have…”

“…However it is possible to apply the indirect limiting procedure to a higher order inequality and another inequality. Indeed, in [29], the indirect limiting procedure is applied to the Rellich inequality, which is known as a higher order generalization of the Hardy inequality, and the Poincaré inequality (For the Rellich inequality, we consider a limit as p ր N 2 . For the Poincaré inequality, we consider a limit as |Ω| ց 0).…”