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

For the kernel $B_{\kappa,a}(x,y)$ of the $(\kappa,a)$-generalized Fourier transform $\mathcal{F}_{\kappa,a}$, acting in $L^{2}(\mathbb{R}^{d})$ with the weight $|x|^{a-2}v_{\kappa}(x)$, where $v_{\kappa}$ is the Dunkl weight, we study the important question of when $\|B_{\kappa,a}\|_{\infty}=B_{\kappa,a}(0,0)=1$. The positive answer was known for $d\ge 2$ and $\frac{2}{a}\in\mathbb{N}$. We investigate the case $d=1$ and $\frac{2}{a}\in\mathbb{N}$. Moreover, we give sufficient conditions on parameters for $\|B_{\kappa,a}\|_{\infty}>1$ to hold with $d\ge 1$ and any $a$. We also study the image of the Schwartz space under the $\mathcal{F}_{\kappa,a}$ transform. In particular, we obtain that $\mathcal{F}_{\kappa,a}(\mathcal{S}(\mathbb{R}^d))=\mathcal{S}(\mathbb{R}^d)$ only if $a=2$. Finally, extending the Dunkl transform, we introduce non-deformed transforms generated by $\mathcal{F}_{\kappa,a}$ and study their main properties.