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 $Q_{\Omega }^{(\alpha )}(t)={\int }_{\Omega }{u}_{\alpha }(x,t)dx$, 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{|Ω|-Q^{(α)}_Ω(t)}{μ_α(t)}=|\partial Ω|_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.

21 pages; Theorem 3.1 has been updated and a different proof has been provided