Fractional heat content asymptotics for Carnot groups

Abstract

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 $Q^{(\alpha)}_\Omega(t)=\int_{\Omega}u_\alpha(x,t) dx$, 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 explicit rate function $\mu_\alpha: (0,\infty)\to (0,\infty)$ such that \begin{align*} \lim_{t\to 0}\frac{|\Omega|-Q^{(\alpha)}_\Omega(t)}{\mu_\alpha(t)}=|\partial \Omega|_H, \end{align*} where $|\Omega|$, $|\partial \Omega|_H$ are the volume and horizontal perimeter of $\Omega$ respectively. Moreover, the rate function $\mu_\alpha$ coincides with the same for the Euclidean case.