Fractional Laplacian and support

Question

Let $u:\mathbb [-1,1] \to \mathbb R$ such that $\mathrm{supp}(u) \subset B_{1/2}(0)$. Under what assumptions on $u$ does it hold $$\mathrm{supp}\Big((-\Delta)^s u\Big) \subset B_{1/2}(0),$$ where $(-\Delta)^s$ denotes the spectral fractional Laplacian?

Answer 1

My bet is that $u$ is necessarily zero. The following may not be fully rigorous, unfortunately I have no time to fill in the details now.

One way to see this is via harmonic extensions and Hopf's lemma. For $s = \tfrac12$, consider the harmonic extension $v(x,y)$ of $u(x)$ to $[-1,1] \times [0,\infty)$, with Neumann condition along the vertical parts of the boundary. Then $v(x,0) = \partial_y v(x,0) = 0$ for $x$ in some interval, which violates the Hopf's lemma. For general $s$ one can give a similar argument using the Caffarelli–Silvestre extension.

Another way to proceed (which only works for intervals in 1-D and for some tiling domains in higher dimensions) is to extend $u$ to a periodic function $\tilde u$ with period $4$, satisfying $u(2 - x) = u(x)$. Then the spectral fractional Laplacian applied to $u$ is equal to the usual fractional Laplacian applied to $\tilde u$, and for the latter the result is well-known, I can look up a reference if you are interested.