Sharp Hardy-Leray inequality for solenoidal fields

TL;DR

This paper determines the optimal constants in Hardy-Leray inequalities for solenoidal vector fields in ^N, extending previous results by removing symmetry assumptions and analyzing the attainability of these constants.

Contribution

It computes the best constants for Hardy-Leray inequalities for general solenoidal fields without symmetry restrictions, improving upon prior axisymmetric results.

Findings

Derived explicit formulas for the best constants.

Showed the best constants are not attained in the function space.

Extended known inequalities to more general solenoidal fields.

Abstract

We compute the best constant in functional integral inequality called the Hardy-Leray inequalities for solenoidal vector fields on $\mathbb{R}^N$. This gives a solenoidal improvement of the inequalities whose best constants are known for unconstrained fields, and develops of the former work by Costin-Maz'ya who found the best constant in the Hardy-Leray inequality for axisymmetric solenoidal fields. We derive the same best constant without any symmetry assumption, whose expression can be simplified in relation to the weight exponent. Moreover, it turns out that the best value cannot be attained in the space of functions satisfying the finiteness of the integrals in the inequality.