Let $p_t(x):=\int_{\mathbb{R}^n} \mathrm{e}^{ix\cdot \xi}\mathrm{e}^{-t|\xi|^2}\,d\xi$ be the heat's kernel. within the properties of the kernel $p_t$, are fulfilled that $p_t(x)>0$ for all $t\geq 0$ (because inverse Fourier transform of gaussian function is again a gaussian function) and $p_t$ is a Schwartz function (because $\mathrm{e}^{-t|\xi|^2}$ is a Schwartz function and Fourier is an automorphism on Schwartz space).
If $P_t(x)=\int_{\mathbb{R}^n}\mathrm{e}^{ix\cdot \xi}\mathrm{e}^{-t (|\xi|^2)^{s/2}}\,d\xi$, $s>0$ (Laplacian fractional's kernel) does the same apply? From what I see, it will depend on the value of each $s$, for example, with $s=1$, $\mathrm{e}^{-t |\xi|}$ is not a Schwartz function because is not differentiable in the origin.