What is $\left\| f \right\|\_{ H\_{0}^{k}, H\_{0}^{k}}$ norm when $H\_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int\_{M} u \operatorname{vol}\_{g}=0\right\}$.


I'm reading a paper
[Chern-Yamabe flow](https://arxiv.org/pdf/1706.04917.pdf)
which said that


**Now for $k$ big enough $(k>n)$, the first eigenvalue of the operator $-\Delta$ on the space $H\_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int\_{M} u \mathrm{vol}\_{g}=0\right\}$ is strictly negative hence $\left\|e^{-t \Delta}\right\|\_{H\_{0}^{k}, H\_{0}^{k}} \leq$ $C\_{0} e^{-c t}$.**


I have no idea what is $\left\| - \right\|\_{ H\_{0}^{k}, H\_{0}^{k}}$ norm.



What is $\left\| u \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$
A partial answer to your question. I would think it means:
$$
\sup\bigg(\| u(t,\cdot)\|\_{H^k\_0} : u(t,\cdot)=\exp(-t\Delta) u(0,\cdot),\textrm{ with } \| u(0,\cdot)\|\_{H^k\_0}=1\bigg).
$$


