Bi-parametric parabolic potentials in the Dunkl-Fourier setting

In this work, we introduce the new notion of bi-parametric parabolic potentials in the framework of the Dunkl-Fourier transform which extends the classical bi-parametric potentials (for $$k=0$$ ). Namely, we define the families of operators $$H^{k}_{\alpha ,\beta }:=\left( \dfrac{\partial }{\partial t}+(-\triangle _k)^{\frac{\beta }{2}}\right) ^{-\frac{\alpha }{\beta }}\,\,\,\,\hbox {and}\,\,\,\, \mathcal{H}^{k}_{\alpha ,\beta }:=\left( I+\dfrac{\partial }{\partial t}+(-\triangle _k)^{\frac{\beta }{2}}\right) ^{-\frac{\alpha }{\beta }}\,\,\,(\alpha ,\,\beta >0),$$ where $$\triangle _k$$ is the Laplace-Dunkl differential operator and I is the identity operator. Also, some properties of these parabolic potentials in the special weighted $$L^p_{k,0}(I\!\!R^d\times I\!\!R)$$ -spaces are established.