I need to check if the different fractional Laplacians have the same properties regarding the semigroups they generate. All I know now is in the case of the Restricted Fractional Laplacian:
$(-\Delta)^s\phi(x)=cP.V.\displaystyle\int_{\mathbb{R}^N} \frac{\phi(x)-\phi(y)}{|x-y|^{N+2s}} dy dx$, $0<s<1$.
Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$.
Now by denoting $A:=-(-\Delta)^s$
with domain $D(A)=$ {$\phi\in\mathbb{H}_{0}^{s}(\Omega),(-\Delta)^s\phi\in L^2(\Omega)$ }.
We have that $A$ generates a strongly continuous semigroup {$S_A(t)$}, which is bounded analytic, contraction and ultracontrative. By ultracontractive, I mean that for any $\phi\in L^p(\Omega)$ and $1\leq p\leq q\leq +\infty$, there exists a constant $C>0$ such that $\lVert S_A(t)\phi \rVert_q\leq C t^{-\frac{N}{2s}(\frac{1}{p}-\frac{1}{q})}\lVert\phi\rVert_p$. Here is a reference presenting all these results see for instance .
What about the Spectral fractional Laplacian? Do we have these properties?
Let us recall two equivalent definitions of the Spectral fractional Laplacian for $0<s<1$:
1- $$
\left(-\Delta_{S p e c}\right)^s \phi(x)=\frac{1}{\Gamma(-s)} \int_0^{+\infty}\left(e^{t \Delta} \phi(x)- \phi(x)\right) \frac{d t}{t^{1+s}},
$$
where $e^{t \Delta} \phi$ represents the solution of the classical heat equation in $\Omega$.
2-$$
\left(-\Delta_{\text {Spec }}\right)^s \phi(x)=\sum_{k \geq 1} \lambda_k^s \phi_k e_k(x),
$$
where $\left\{e_k, \lambda_k\right\}_{k \in \mathbb{N}^*}$ are the eigenfunctions and the eigenvalues of $-\Delta$ in $\Omega$ with the homogeneous Dirichlet boundary condition, and $\phi_k$ represents the projection of $\phi$ in the direction $e_k$.
Is the semigroup generated by the Spectral fractional Laplacian strongly continuous, bounded analytic, contraction and ultracontrative? Would you please help me to find appropriate references in this regard?
Thank you very much in advance.
Best Answer
Yes on all counts. The first question should be whether the spectral fractional Laplacian generates a semigroup at all. This is indeed the case. An explicit formula is $$ P_t f=\sum_{k=1}^\infty e^{-t\lambda_k^s}\langle e_k,f\rangle e_k, $$ where the series converges in $L^2(\Omega)$. A direct computation shows that the spectral fractional Laplacian is indeed the generator. Since it is densely defined, the semigroup is strongly continuous, and since it is positive self-adjoint on $L^2(\Omega)$, the semigroup is bounded analytic and contractive. All this can be read off directly from the formula above.
The ultracontractivity (with the same rate function as for the restricted fractional Laplacian) follows from the equivalent characterization in terms of Sobolev inequalities, see Cowling, Meta: Harmonic Analysis and Ultracontractivity, Transactions of the American Mathematical Society, 1993 for example.