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

We consider an integral transform given by $T_{\nu }f(s):=\pi {\int }_0^{\mathrm{\infty }}rs{J}_{\nu }(rs{)}^{2}f(r)dr$, 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 equations 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 $x\mapsto f(\mid x\mid )$ when $\nu =k+d\mathrm{/}2-1$ for $k\in \mathbb{N}$. The aims of this paper are to extend their identity for non-integer indices and to derive several inequalities from it.