Sharp Hardy-Leray inequality for curl-free fields with a remainder term

In this paper, we give a new and a simpler approach to the result in [8] concerning the best constant of Hardy-Leray inequality for curl-free fields. As a by-product, we obtain an improved inequality with a remainder term. The non-attainability of the best constant is an easy consequence of the new inequality. The proof is based on a decomposition of curl-free fields into radial and spherical parts.