Identities and inequalities for integral transforms involving squares of the Bessel functions
Soichiro Suzuki
TL;DR
This work analyzes the transform $T_{\nu} f(s) = \pi \int_0^\infty rs J_{\nu}(rs)^2 f(r)\,dr$, which arises in sharp smoothing estimates for the free Schrödinger equation. It establishes a non-integer extension of an identity linking $T_{\mu+\nu}$ to a Hankel-transform operator $U_{\mu,\nu}$ and the $\nu$-Hankel transform, and derives a suite of kernel-based inequalities by comparing Gegenbauer-extended kernels. The authors show that for completely monotone radial data (via Gaussian mixtures), $T_{\nu} f$ enjoys complete monotonicity in a scaled variable and monotonicity properties in $\nu$, with explicit forms such as $T_{\nu} \varphi(s) = \pi s e^{-s^2} I_{\nu}(s^2)$ for $\varphi(r)=e^{-r^2/2}$. Extensions to ν = -1/2 and to Dirac-type settings are discussed, along with implications that one inequality entails another, and the results provide a robust framework for understanding dimension-dependent smoothing constants in dispersive PDEs.
Abstract
We consider an integral transform given by $T_ν f(s) := π\int_0^\infty rs J_ν(r s)^2 f(r) \, dr$, where $J_ν$ denotes the Bessel function of the first kind of order $ν$. 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 $T_ν f$ involving the $d$-dimensional Fourier transform of $x \mapsto f(\lvert x \rvert)$ when $ν= k + d/2 - 1$ 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.