\nu}=0 ,&\text{in }{{\mathbb R}}^{N},\\
\frac{\partial V}{\partial
  \nu}=h\, U(\cdot,0),&\text{in }{{\mathbb R}}^{N},
\end{cases}$$ which arise naturally as extension of fractional equations of the form $$(-\Delta)^{3/2}u=a(x)u$$ once we put $h=-K_{3/2}^{-1}\, a=-2a$. Indeed, by [@FF-prep Proof of Lemma 3.2, Step 6] we deduce that the constant $C_b$ defined there equals $\sqrt 2$ when $b=0$ and, since it can be shown that $K_s=C_b^{-2}$ with $b=3-2s$, we deduce that $K_{3/2}=\frac 12$. The frequency function associated to problem is given by the ratio of the local energy over mass near the fixed point $0\in{{\mathbb R}}^N$ $$\label{eq:59}
\mathcal N(r)=\frac{r^{-N+1} \left[\int_{B_r^+} \left(|\nabla U|^2+|\nabla V|^2+UV\right)dz -\int_{B_r'} h(x)U(x,0)V(x,0)\, dx\right]}{r^{-N}\int_{S_r^+} (U^2+V^2)\, dS},$$ where we are denoting as $z=(x,t)\in {{\mathbb R}}^N\times{{\mathbb R}}$ the variable in ${{\mathbb R}}^{N+1}={{\mathbb R}}^N\times{{\mathbb R}}$, $dz=dx\,dt$ and, for all $r>0$, $$\begin{aligned}
&B_r=\{z\in {{\mathbb R}}^{N+1}:|z|<r\},\quad
B_r^+=\{(x,t)\in B_r:t>0\},\\
&B_r'=\{x\in {{\mathbb R}}^{N}:|x|<r\}=B_r\cap({{\mathbb R}}^N\times\{(x,0):x\in{{\mathbb R}}^N\}),\\
&S_r^+=\{(x,t)\in \partial B_r:t>0\}.\end{aligned}$$ The classical approach developed by Garofalo and Lin [@GL] to prove unique continuation through Almgren’s monotonicity formula is based on the validity of doubling type conditions, obtained as a consequence of boundedness of the quotient $\mathcal N$. We refer to [@ae; @aek; @FFFN; @FF-edinb; @KukavicaNystrom; @tz1; @tz2] for unique continuation from the boundary established via Almgren monotonicity formula.

While in the local case doubling conditions 