We propose a novel approach for studying small-time asymptotics of the fractional heat content of $C^2$ non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by $\mathcal{ L}$, the fractional heat content of a bounded domain $\Omega$ is defined as $Q^{ (\alpha)}_\Omega(t)=\int_{ \Omega} u_\alpha(t,x)dx$, where $u_\alpha$ is the solution to the heat equation corresponding to the fractional sub-Laplacian $\mathcal{ L}_\alpha:=\mathcal{ L}^{ \alpha/2}$ with Dirichlet boundary condition on $\Omega$. We prove that for $1\le\alpha\le 2$, there exists explict rate function $\mu_\alpha: (0,\infty)\to (0,\infty)$ such that \[ \lim_{ t\to 0}\frac{ |\Omega|-Q^{ (\alpha)}_\Omega(t)}{ \mu_\alpha(t)}=|\partial\Omega|_H, \] where $|\partial\Omega|_H$ is the horizontal perimeter of $\Omega$. Moreover, the rate function $\mu_\alpha$ coincides with the same for the Euclidean case.
Categories: math.AP math.PR