# Does an integration by parts formula hold for the spectral fractional Laplacian in 1-d?

Is there an integration by parts formula for the spectral fractional Laplacian in a bounded interval $$[a,b] \subset \mathbb R$$?

• What do you mean by integration by parts? Something like $\int_a^b f(x) (-\Delta_\Omega)^s g(x) dx = \int_a^b g(x) (-\Delta_\Omega)^s f(x) dx$? This is obviously true for $f$ and $g$ in the $L^2$ domain of $(-\Delta_\Omega)^s$, as this is a self-adjoint operator (at least as long as $\Delta_\Omega$ is self-adjoint, so we put, say, Dirichlet boundary condition at $\partial \Omega$). On the other hand, for more general $f, g$ it is not clear what one would understand by $(-\Delta_\Omega)^s f(x)$. – Mateusz Kwaśnicki Jun 28 at 23:15