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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:

  1. 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.)
  2. 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?
  3. 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.

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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.

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