Remarks on a limiting case of Hardy type inequalities

TL;DR

This paper explains the emergence of the logarithmic function in the critical Hardy inequality as a limiting case of the classical Hardy inequality when the parameter approaches the dimension, and extends the approach to related inequalities.

Contribution

It derives the logarithmic correction in the critical Hardy inequality from the classical inequality via a limiting process, and applies this method to Rellich and Poincaré inequalities.

Findings

Logarithmic function arises as a limit of Hardy potential when p approaches N.

Limiting procedure applies to Rellich and Poincaré inequalities.

Provides a new perspective on the critical Hardy inequality derivation.

Abstract

The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. 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 $W_0^{1,N}$. 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 $p \nearrow N$. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces $W_0^{2,p}$ with $p \in (1, \frac{N}{2})$ and the Poincar\'e inequality.