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I am wondering whether it is possible to define, by duality, the fractional Laplacian of a measure $\rho$, for $0<s<1$, let's say on the whole space $\mathbf{R}^d$. I would define it as a distribution $(-\Delta)^s(\rho) \in \mathcal{D}'(\mathbf{R}^d)$ given by: $$ \left\langle (-\Delta)^s(\rho), \varphi \right\rangle := \int (-\Delta)^s \varphi d \rho, $$ which would make sense since $(-\Delta)^s \varphi \in \mathcal{C}^\infty \cap L^\infty$... After searching for it, i didn't find any sort of definition, so I believe I made a naive mistake somewhere.

The point would be to consider measure-valued solutions of a PDE involving a fractional Laplacian.

Sugiton
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    $-\Delta^s\phi$ is a Schwartz function, so you need to assume some decay condition on $\rho$. But you could let $\rho$ be any tempered distribution, and your definition works. – Mason Nov 21 '22 at 17:06
  • Thank you, that answers my question :) – Sugiton Nov 27 '22 at 17:35

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