Sharp Rellich-Leray inequality for axisymmetric divergence-free vector fields

In this paper, we show the N-dimensional Rellich-Leray inequality with optimal constant for axisymmetric and divergence-free vector fields. This is a second-order differential version of the former work by Costin-Maz’ya (Calc Var Partial Differ Equ 32(4):523–532, 2008) on sharp Hardy–Leray inequality for such vector fields. In the proof of our main theorem, we show the vanishing of azimuthal components of axisymmetric vector fields for N≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 4$$\end{document}, from which we also find a partial modification of the best constant derived in Costin-Maz’ya (Calc Var Partial Differ Equ 32(4):523–532, 2008).