Sharp Hardy-Leray inequality for solenoidal fields

Abstract

We compute the best value of the constant in functional integral inequality called the Hardy-Leray inequality for solenoidal vector fields on RN. This gives a solenoidal improvement of the original best constant known for unconstrained vector fields, and refines the former work by Costin-Maz'ya [4] who found a new best constant in Hardy-Leray inequality for axisymmetric solenoidal vector fields. Our method does not require any symmetry assumption to derive the same constant number as Costin-Maz'ya's. Moreover, the constant has a simpler expression in relation to the weight exponent and turns out to be unattainable in the space of functions satisfying some regularity condition fitting to the Hardy-Leray inequality.