In this paper, following ^[6], we study the regularity properties of Bessel potentials and Bessel fractional derivatives in the context of variable Gaussian Besov $-$Lipschitz spaces $B_{p(\cdot ),q(\cdot )}^{\alpha }({\gamma }_d),$ which were defined and studied in ^[9], under certain conditions on $p(\cdot )$ and $q(\cdot )$.

In this paper, we introduce variable Gaussian Besov-Lipschitz $B_{p(\cdot ),q(\cdot )}^{\alpha }({\gamma }_d)$ and Triebel-Lizorkin spaces $F_{p(\cdot ),q(\cdot )}^{\alpha }({\gamma }_d),$ i.e., Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces with variable exponents $p(\cdot )$ and $q(\cdot )$, under certain regularity conditions on the functions $p(\cdot )$ and $q(\cdot )$. The condition on $p(\cdot )$ is associated with the Gaussian measure and was introduced in ^[3]. Trivially, they include the Gaussian Besov-Lipschitz $B_{p,q}^{\alpha }({\gamma }_d)$ and Triebel-Lizorkin spaces $F_{p,q}^{\alpha }({\gamma }_d)$ for $p,q$ constants, which were introduced and studied in ^[10]. We consider some inclusion relations of those spaces and finally prove some interpolation results for them.