An order-interpolation inequality for Bessel functions

Abstract

We show that $J_{\hat{I}¼+\hat{I}½}(r{)}^2<J_{\hat{I}½-1/2}(r{)}^2+J_{\hat{I}½+1/2}(r{)}^2$ holds whenever $\hat{I}¼\in (-1/2,1/2)$, $\hat{I}½\in [0,\infty )$, and $r\in (0,\infty )$. In fact, we prove a stronger version for any fixed non-trivial linear combination of the Bessel functions of the first and second kinds. This inequality can be regarded as a kind of interpolation with respect to order. As an application, we establish a dimension-comparison result for optimal constants of smoothing estimates for the free Schrödinger equation. Briefly, the optimal constant on ${\mathbb{R}}^{d+1}$ is at most twice that on ${\mathbb{R}}^d$ for each $d\ge 2$.

7 pages, 1 figure