Bounds for the kernel of the $(\kappa ,a)$-generalized Fourier transform

In this paper, we study the pointwise bounds for the kernel of the $(\kappa ,a)$-generalized Fourier transform with $\kappa \equiv 0$, introduced by Ben Saïd, Kobayashi and Orsted. We present explicit formulas for the case $a=4$, which show that the kernels can exhibit polynomial growth. Subsequently, we provide a polynomial bound for the even dimensional kernel for this transform, focusing on the cases with finite order. Furthermore, by utilizing an estimation for the Prabhakar function, it is found that the $(0,a)$-generalized Fourier kernel is bounded by a constant when $a>1$ and $m\ge 2$, except within an angular domain that diminishes as $a\to \mathrm{\infty }$. As a byproduct, we prove that the $(0,2^{\mathrm{\ell }}\mathrm{/}n)$-generalized Fourier kernel is uniformly bounded, when $m=2$ and $\mathrm{\ell },n\in \mathbb{N}$.