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?
1 Answer
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 1D 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 wellknown, I can look up a reference if you are interested.

$\begingroup$ Thanks! Yes, what is the reference? If the result is not true, can we estimate the size of $u$ outside the ball? $\endgroup$– LaoCommented Mar 31, 2021 at 11:12

$\begingroup$ What about the case where $u$ decays well on the support; for example, $u \sim x^{\alpha}$, with $\alpha$ big enough near the support? $\endgroup$– LaoCommented Mar 31, 2021 at 11:26

1$\begingroup$ One reference is Fall–Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. PDE 39(2) (2014): 354–397, DOI:10.1080/03605302.2013.825918. This uses the harmonic extension technique, but I remember seeing this in an earlier work, proved with direct methods. I think Krylov's paper DOI:10.1016/j.jfa.2019.02.012 is something one should check. Regarding estimating the size of $u$ or $(\Delta)^s u$ — I do not know. $\endgroup$ Commented Mar 31, 2021 at 11:29