On Hardy inequalities with a remainder term

TL;DR

This paper investigates improved Hardy inequalities by adding Lorentz norm-dependent remainder terms, determining optimal constants, and reducing the problem to a spherically symmetric case, especially novel for gradient-based norms.

Contribution

It introduces new Hardy inequalities with Lorentz norm-based remainders and establishes the optimal constants by simplifying to symmetric cases, including the gradient norm scenario.

Findings

Optimal constants for Hardy inequalities with Lorentz norms identified.

Reduction to spherically symmetric problems simplifies the analysis.

New results for inequalities involving Lorentz norms of gradients.

Abstract

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of $u$ or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. This result is new when the right hand side is a Lorentz norm of the gradient.