Let $A=-(-\Delta)^s$ with domain $D(A)=$ {$\phi\in\mathbb{H}_{0}^{s}(\Omega),(-\Delta)^s\phi\in L^2(\Omega)$ }. $(-\Delta)^s$ is the fractional Laplacian of order $s\in(0,1)$. We know that $A$ generates a strongly continuous semigroup {$S_A(t)$}, see for instance .
There is also a know result about the ultracontrativity of $A$. Namely, for any $\phi\in L^p(\Omega)$ and $1\leq p\leq q\leq +\infty$, there exists a constant $C>0$ such that $\lVert S_A(t)\phi \rVert_q\leq C t^{-\frac{N}{2s}(\frac{1}{p}-\frac{1}{q})}\lVert\phi\rVert_p$,
where $\Vert \cdot \Vert_p = \Vert \cdot \Vert_{L^p(\Omega)}$.
I think the proof of the this estimate follows from the fact that $A$ is a non-negative self-adjoint operator on $L^2(\Omega)$ with compact resolvent and that the semigroup ${S_A(t)}_{t\geq 0}$ is submarkovian. Would you please help me to find a reference that proves the ultracontractivity whether in the case of fractional Laplacian or in general case (for self-adjoint operators with Dirichlet condition). Thank you very much in advance.