Rellich Inequality via Radial Dissipativity

Abstract

We give a conceptually simple and essentially one-dimensional approach to Rellich inequality in Euclidean space ${\mathbb{R}}^n$. In particular, we show that the radial part and the spherical part of the standard Laplacian form an angle in $[0,\pi /2]$ when $n\ge 4$, a property known in the works of Evans-Lewis, Machihara-Ozawa-Wadade, and Bez-Machihara-Ozawa. Our proof here is direct and short.