\text{on} \
{{\mathbb R}}^N\times\{0\}
\right\}$$ with respect to the norm $$\|U\|_{\mathcal D}=\bigg(\int_{{{\mathbb R}}^{N+1}_+} |\Delta
U(x,t)|^2\,dx\,dt\bigg)^{\!\!1/2}.$$ By [@FF-prep] there exists a well defined continuous trace map $$\mathop{\rm Tr}:\mathcal D\to \mathcal D^{3/2,2}({{\mathbb R}}^N),$$ where the space $\mathcal D^{3/2,2}({{\mathbb R}}^N)$ is defined as the completion of $C^\infty_c({{\mathbb R}}^N)$ with respect to the scalar product $$\label{eq:scalar-product}
(u,v)_{\mathcal D^{3/2,2}({{\mathbb R}}^N)}:=\int_{{{\mathbb R}}^N} |\xi|^{3}\, \widehat
u(\xi)\overline{\widehat v(\xi)} \, d\xi.$$ In $\widehat u$ denotes the Fourier transform of $u$ in ${{\mathbb R}}^N$: $$\widehat u(\xi)=\frac 1{(2\pi)^{N/2}} \int_{{{\mathbb R}}^N} e^{-i\xi x}u(x)\, dx \, .$$ Moreover in we denoted by $\overline{\widehat v(\xi)}$ the complex conjugate of $\widehat v(\xi)$.

We observe that, since $u$ and $v$ are real functions, is really a scalar product although their respective Fourier transforms are complex functions.

As a corollary of Theorem \[t:asym\] we derive sharp asymptotic estimates and a strong unique continuation principle for weak $\mathcal D$-solutions to the fourth order elliptic problem $$\label{eq:44}
  \begin{cases}
    \Delta^2 U=0,&\text{in }{{\mathbb R}}^{N+1}_+,\\[3pt]
\frac{\partial U}{\partial\nu}=0,&\text{in }{{\mathbb R}}^N \, , \\[3pt]
\frac{\partial(\Delta U)}{\partial\nu}=h\, \mathop{\rm
  Tr}(U),&\text{in }\Omega \, .
  \end{cases}$$ By a weak $\mathcal D$-solution to we mean some $U\in
\mathcal D$ such that $$\int_{{{\mathbb R}}^{N+1}_+}\Delta U(x,t) \Delta \varphi(x,t)\,dx\,dt=-\int_\Omega
h(x) \mathop{\rm Tr}U(x)
\mathop{\rm Tr}\varphi(x)\,dx$$ for all $\varphi\in\mathcal D$ such that $\mathop{\rm supp}(\mathop{\rm 