Fractional heat content asymptotics for Carnot groups
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909008938401792 |
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| author | Sarkar, Rohan |
| author_facet | Sarkar, Rohan |
| contents | 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 $Ω$ is defined as $Q^{(α)}_Ω(t)=\int_Ωu_α(x,t) dx$, where $u_α$ is the solution to the heat equation corresponding to the fractional sub-Laplacian $\mathcal{L}_α:=\mathcal{L}^{α/2}$ with Dirichlet boundary condition on $Ω$. We prove that for $1\le α\le 2$, there exists explicit rate function $μ_α: (0,\infty)\to (0,\infty)$ such that
\begin{align*}
\lim_{t\to 0}\frac{|Ω|-Q^{(α)}_Ω(t)}{μ_α(t)}=|\partial Ω|_H,
\end{align*}
where $|Ω|$, $|\partial Ω|_H$ are the volume and horizontal perimeter of $Ω$ respectively. Moreover, the rate function $μ_α$ coincides with the same for the Euclidean case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04088 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractional heat content asymptotics for Carnot groups Sarkar, Rohan Analysis of PDEs Probability 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 $Ω$ is defined as $Q^{(α)}_Ω(t)=\int_Ωu_α(x,t) dx$, where $u_α$ is the solution to the heat equation corresponding to the fractional sub-Laplacian $\mathcal{L}_α:=\mathcal{L}^{α/2}$ with Dirichlet boundary condition on $Ω$. We prove that for $1\le α\le 2$, there exists explicit rate function $μ_α: (0,\infty)\to (0,\infty)$ such that \begin{align*} \lim_{t\to 0}\frac{|Ω|-Q^{(α)}_Ω(t)}{μ_α(t)}=|\partial Ω|_H, \end{align*} where $|Ω|$, $|\partial Ω|_H$ are the volume and horizontal perimeter of $Ω$ respectively. Moreover, the rate function $μ_α$ coincides with the same for the Euclidean case. |
| title | Fractional heat content asymptotics for Carnot groups |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2601.04088 |