Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition：
\begin{equation}
\left\{
\begin{array}{ll}
\partial_{t}u(x,t)+(-\triangle)^{s}u(x,t)=0 , & \hbox{in $\Omega$, $t>0$;} \\
u(x,t)=0, & \hbox{in~$\mathbb{R}^n\setminus\Omega, t\geq 0$;} \\
u(x,0)=f_{0}(x)\in L^2(\Omega), & \hbox{in $\Omega$, for $t=0$.}
\end{array}
\right.
\end{equation}
Here $s\in (0,1)$, $(-\triangle)^{s}$ is the fractional Laplacian given by
\begin{equation}
(-\triangle)^{s}u(x)=\mathcal{F}^{-1}(|\xi|^{2s}\hat{u}(\xi))(x),
\end{equation}
with $\hat{u}(\xi)=\mathcal{F}u(\xi)=(2\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^n}u(x)e^{-ix\cdot
\xi}dx$ is the Fourier transform of $u$. In paper [1], the author presented a Dirichlet kernel $h_{D}(x,y,t)$ of $(-\triangle)^{s}$ on $\Omega$ and a global heat kernel $h(x,y,t)$ of $(-\triangle)^{s}$ in $\mathbb{R}^n$. Then, the author claimed that (page 221 in [1]) we can deduce from the maximum principle that

$$ 0\leq h_{D}(x,y,t)\leq h(x,y,t)=\int_{\mathbb{R}^n}e^{-t|\xi|^{2s}}e^{i \xi\cdot (x-y)}\frac{d\xi}{(2\pi)^{n}}~~\mbox{for all}~x,y\in \Omega. $$
Then, we have
$$ \sum_{j=1}^{+\infty}e^{-t\lambda_{j}}|\phi_{j}(x)|^{2}\leq \frac{\omega_{n}}{(2\pi)^{n}}\Gamma\left(1+\frac{n}{2s}\right)t^{-\frac{n}{2s}}~~\mbox{for all}~~x\in \Omega,$$
where $\omega_{n}$ is the Lebesgue of unit ball in $\mathbb{R}^n$, $\lambda_{j}$ and $\phi_{j}$ are denoted by the $j$-th Dirichlet eigenvalue and Dirichlet eigenfunction of $(-\triangle)^{s}$ on $\Omega$.

**Here is my Question:**

- I feel very confused about how can we deduce that $h_{D}(x,y,t)\leq h(x,y,t)$ by the maximum principle of fractional Laplacian. Because the regularity results and basic properites of fractional Dirichlet kernel $h_{D}(x,y,t)$ are not clear for me. How can we deduce that $h_{D}(x,y,t)\leq h(x,y,t)$ by the maximum principle? What is the maximum principle for the fractional heat equation? (I found some versions of maximum principles in [4], but it seems not worked.)
- Since the fractional Laplacian $(-\triangle)^{s}$ is a non-local operator, it seems we cannot use the classical approach as classical Laplacian to establish the Dirichlet kernel. I found some papers such as [2] [3]，but all of them use the symmetric $\alpha$-stable process to establish the fractional Dirichlet heat kernel for $(-\triangle)^{s}$, and little PDE properties was involved. Can someone give an approach (or a detail reference) to establish the fractional Dirichlet heat kernel in view of classical PDE sense? What is the complete definition of fractional Dirichlet heat kernel?
- Can we claim that the series $$ h_{D}(x,y,t)=\sum_{j=1}^{\infty}e^{-\lambda_{j}t}\phi_{j}(x)\phi_{j}(y)$$ converges uniformly on $\overline{\Omega}\times\overline{\Omega}\times [\varepsilon,+\infty)$ for any $\varepsilon>0$?

For the third question, it seems we can deduce from the fractional Sobolev embedding inequality (cf. [4] and [5]) that $\|\phi_{k}\|_{L^{\infty}(\Omega)}\leq C\cdot \lambda_{k}^{\frac{n}{4s}}$ and from [4] we know the eigenfunctions $\phi_{k}\in C^{\infty}(\Omega)\cap C^{s}(\overline{\Omega})$.

Can someone help me? Thank you very much!

Reference:

[1] *Frank, Rupert L.*, **Eigenvalue bounds for the fractional Laplacian: a review**, Palatucci, Giampiero (ed.) et al., Recent developments in nonlocal theory. Berlin: De Gruyter Open (ISBN 978-3-11-057155-4/hbk; 978-3-11-057156-1/ebook). 210-235 (2018). ZBL1404.35303.

[2] *Bañuelos, Rodrigo; Kulczycki, Tadeusz; Siudeja, Bartłomiej*, **On the trace of symmetric stable processes on Lipschitz domains**, J. Funct. Anal. 257, No. 10, 3329-3352 (2009). ZBL1189.60100.

[3] *Chen, Zhen-Qing; Kim, Panki; Song, Renming*, **Heat kernel estimates for the Dirichlet fractional Laplacian**, J. Eur. Math. Soc. (JEMS) 12, No. 5, 1307-1329 (2010). ZBL1203.60114.

[4] *Fernández-Real, Xavier; Ros-Oton, Xavier*, **Boundary regularity for the fractional heat equation**, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 110, No. 1, 49-64 (2016). ZBL1334.35386.

[5] *Brasco, L.; Lindgren, Erik; Parini, Enea*, **The fractional Cheeger problem**, Interfaces Free Bound. 16, No. 3, 419-458 (2014). ZBL1301.49115.