Some properties of fractional Dirichlet heat kernel 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.
 A: The answer to your question 3 is affirmative: this follows from ultracontractivity of the heat operators.
In fact, even more is true: the series $$\frac{1}{\phi_1(x) \phi_1(y)} \sum e^{-t \lambda_j} \phi_j(x) \phi_j(y)$$ converges uniformly to $h_\Omega(t,x,y)/(\phi_1(x) \phi_1(y))$ on $[\epsilon, \infty) \times \Omega \times \Omega$ with no regularity assumptions on $\Omega$, other than it has finite Lebesgue measure. This follows from intrinsic ultracontractivity of the heat semigroup, a result proved independently by Tadeusz Kulczycki (for general domains), and Zhen-Qing Chen and Renming Song (for smooth domains, I believe) in 1997–98.
Ultracontractivity asserts that the heat operator $H(t)$ is bounded from $L^2(\Omega)$ to $L^\infty(\Omega)$. Since the series $$h_\Omega(t, x, \cdot) = \sum e^{-t \lambda_j} \phi_j(x) \phi_j(\cdot)$$ converges in $L^2(\Omega)$ for $t = \epsilon/2$, uniformly with respect to $x$, applying $H(\epsilon/2)$ to both sides proves that the above series converges uniformly with respect to $(x, y) \in \Omega \times \Omega$ for $t = \epsilon$. Another application of $H(t-\epsilon)$ extends this to all $t \in [\epsilon, \infty)$.
Intrinsic ultracontractivity works as above, but for the intrinsic semigroup, with kernel $h_\Omega(t,x,y) / (\phi_1(x) \phi_1(y))$, and with $L^2(\Omega)$ replaced by the weighted space $L^2((\phi_1(x))^2 dx)$.

Unfortunately I cannot help much with regard to questions 1 and 2: probability is my mother tongue.
