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We can define the spectral power of the Dirichlet Laplacian $(-\Delta_\Omega)$ on $\Omega$. … Then, on the whole space, the fractional laplacian $(-\Delta)^s$ is an operator defined via Fourier transform: $$\mathcal F((-\Delta)^s u)(\xi):= |\xi|^{2s} \mathcal F (u)(\xi)\, .$$ Then, do we have the …

On the whole space $\mathbb R^d$, the fractional Sobolev space $H_s(\mathbb R^d)$ of order $s\in \mathbb R$ can be defined as the subspace of tempered distributions $T$ such that $\mathcal F T \in L^ …

set of functions $u\in L^2(\Omega)$ such that $u=\sum_{k\geq 1}u_k \Phi_k$, and $\sum_{k\geq 1}\lambda_k^s u_k^2 <+\infty$, where $\lambda_k$ and $\Phi_k$ are the eigenvalues and eigenfunctions of the Laplacian …

that for $s\in [0,1]$, $\left [ H_0^1(\Omega), L^2(\Omega)\right ]_\theta=\text{dom}(-\Delta)^{\frac s 2}$, where $\Omega$ is a smooth and bounded domain in $\mathbb R^d$, $(-\Delta)$ is the Dirichlet Laplacian …