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.
Categories: math.CA