# Fractional Laplacian and support

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?

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.

• Thanks! Yes, what is the reference? If the result is not true, can we estimate the size of $u$ outside the ball?
– Lao
Commented Mar 31, 2021 at 11:12
• What about the case where $u$ decays well on the support; for example, $u \sim x^{\alpha}$, with $\alpha$ big enough near the support?
– Lao
Commented Mar 31, 2021 at 11:26
• 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. Commented Mar 31, 2021 at 11:29