Let $u,v \in C^\infty(\Omega)$ and assume that $v$ is compactly supported inside a domain $\Omega$. Is it true that $$ \int_\Omega v (-\Delta)^su \, d x = \int_\Omega (-\Delta)^{s/2}v(-\Delta)^{s/2}u \, dx, $$ where $(-\Delta)^s$ denotes the spectral fractional Laplacian with either Dirichlet or Neumann bounary conditions? Moreover, is it true that $$ \int_K v (-\Delta)^su \, d x = \int_K (-\Delta)^{s/2}v(-\Delta)^{s/2}u \, dx, $$ if $\mathrm{supp}(v) \subseteq K \subset \Omega$?
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