R}}^N)}=\int_{\Omega}a\,u\,\varphi\,dx\quad\text{for all }
\varphi\in C^\infty_c(\Omega).$$ Let us also assume that $$\label{eq:53}
  (-\Delta)^{3/2}u\in (\mathcal D^{1/2,2}({{\mathbb R}}^N))^\star,$$ where $(\mathcal D^{1/2,2}({{\mathbb R}}^N))^\star$ denotes the dual space of $\mathcal D^{1/2,2}({{\mathbb R}}^N)$, in the sense that the linear functional $\varphi\mapsto \int_{{{\mathbb R}}^N} |\xi|^{3} \widehat
  u(\xi)\overline{\widehat \varphi(\xi)} \, d\xi$, $\varphi\in C^\infty_c({{\mathbb R}}^N)$, is continuous with respect to the norm induced by $\mathcal D^{1/2,2}({{\mathbb R}}^N)$.

1.  If $u$ vanishes at some point $x_0\in\Omega$ of infinite order, i.e. if $$\label{eq:47}
    u(x)=o(|x-x_0|^n)\quad\text{as }x\to x_0\text{ for every }n\in{{\mathbb N}},$$ then $u\equiv 0$ in $\Omega$.

2.  If $u$ vanishes on a set $E\subset\Omega$ of positive Lebesgue measure, then $u\equiv 0$ in $\Omega$.

We observe that assumption is satisfied in each of the following cases:

(i) $u\in \mathcal D^{5/2,2}({{\mathbb R}}^N)$;

(ii) $u\in \mathcal D^{3/2,2}({{\mathbb R}}^N)$ solves with $\Omega={{\mathbb R}}^N$ and $a\in L^{N/2}({{\mathbb R}}^N)\cap C^1({{\mathbb R}}^N)$.

The proof of Theorem \[t:frac\] is based on Theorem \[t:asym-ho\] and the generalization of the Caffarelli-Silvestre extension to higher order fractional laplacians given in [@Y], see also [@FF-prep]. Indeed, according to [@Y], we have that if $u$ solves , then $u$ is the trace on ${{\mathbb R}}^N\times\{0\}$ of some $U\in\mathcal D$ solving with $h=-2a$.

We observe that the unique continuation result stated in Theorem \[t:frac\] does not 