Best constant of the critical Hardy-Leray inequality for curl-free fields in two dimensions

Abstract

. In this note, we prove that the best-possible constant of the critical Hardy-Leray in- equality for curl-free fi elds is 1 / 4, just the same value as the one for all smooth fi elds. This fact contrasts sharply with the recent result on the subcritical Hardy-Leray inequality for curl-free fi elds by the authors [6], and shows the criticality of the inequality.