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

We consider an integral transform given by $T_{\nu} f(s) := \pi \int_0^\infty rs J_{\nu}(r s)^2 f(r) \, dr$, 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 Schr\"{o}dinger 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 $x \mapsto f(\lvert x \rvert)$ when $\nu = k + d/2 - 1$ 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.