Boundedness of the Gaussian Riesz Potentials on Gaussian variable Lebesgue spaces

In this paper we prove the boundedness of the Gaussian Riesz potentials $I_{\beta }$, for $\beta \ge 1$ on $L^{p(\cdot )}({\gamma }_d)$, the Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot )$ following \cite{DalSco}. Additionally, this result trivially gives us an alternative proof of the boundedness of Gaussian Riesz potentials $I_{\beta }$ on Gaussian Lebesgue spaces $L^p({\gamma }_d)$.