A logarithmic Hardy inequality

TL;DR

This paper introduces a new scale-invariant inequality that improves the Hardy inequality by controlling a nonlinear integral with super-quadratic growth using the energy, with explicit sharp constants and minimizers in the radial case.

Contribution

It establishes a novel Hardy-type inequality involving a nonlinear integral with super-quadratic growth, extending classical results and providing explicit sharp constants and minimizers.

Findings

Derived explicit sharp constants for the inequality.

Identified minimizers in the radial case.

Showed the inequality's scale invariance and non-achievability of sharp constants without symmetry.

Abstract

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved among radial functions, in some range of the parameters.