Identities and inequalities for integral transforms involving squares of the Bessel functions

Abstract

We consider an integral transform given by $T_{\nu }f(s):=\pi {\int }_0^{\infty }rs{J}_{\nu }(rs{)}^{2}f(r)dr$, where $J_{\nu }$ denotes the Bessel function of the first kind of order $\nu$. As shown by Walther (2002, doi:10.1006/jfan.2001.3863), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schrödinger equation on ${\mathbb{R}}^d$. On the other hand, Bez et al. (2015, doi:10.1016/j.aim.2015.08.025) studied these optimal constants using a different method, and obtained a certain alternative expression for $T_{\nu }f$ involving the $d$-dimensional Fourier transform of $x\mapsto f(|x|)$ when $\nu =k+d/2-1$ for $k\in \mathbb{N}$. In this paper, we extend their identity to non-integer indices and derive several inequalities from it. In particular, we give a dimension-comparison result for the smoothing estimates on ${\mathbb{R}}^d$ and ${\mathbb{R}}^{d+1}$.

20 pages. Revised according to the referee's report