to the norm $$\|U\|_{\mathcal D^{1,2}({{\mathbb R}}^{N+1}_+)}:=\left(\int_{{{\mathbb R}}^{N+1}_+}
|\nabla U|^2 dz\right)^{1/2} \qquad \text{for any } U\in
C^\infty_c(\overline{{{\mathbb R}}^{N+1}_+}) \, .$$ From [@BCdPS], we have that there exists a constant $K>0$ such that $$\label{eq:C-b}
K \|{\rm Tr\, }U\|_{\mathcal D^{\frac12,2}({{\mathbb R}}^N)}{\leqslant}\|U\|_{\mathcal
D^{1,2}({{\mathbb R}}^{N+1}_+)} \qquad \text{for any } U\in \mathcal
D^{1,2}({{\mathbb R}}^{N+1}_+)  .$$ Here we are denoting as ${\rm Tr\, }$ the trace operator ${\rm Tr\, }:\mathcal D^{1,2}({{\mathbb R}}^{N+1}_+)\to \mathcal D^{\frac12,2}({{\mathbb R}}^N)$. We recall that, for all $\gamma<\frac N2$ the following Sobolev embedding holds: there exists a positive constant $S(N,\gamma)$ depending only on $N$ and $\gamma$ such that $$\label{eq:Sobolev-frazionario}
S(N,\gamma) \|u\|_{L^{2^*(N,\gamma)}({{\mathbb R}}^N)}^2{\leqslant}\|u\|_{\mathcal
D^{\gamma,2}({{\mathbb R}}^{N})}^2 \qquad \text{for any } u\in
C^\infty_c({{\mathbb R}}^N)$$ where $2^*(N,\gamma)=2N/(N-2\gamma)$. Moreover the following Hardy type inequality due to Herbst [@herbst] holds: there exists $\Lambda>0$ $$\label{eq:frac_hardy}
  \Lambda\int_{{{\mathbb R}}^N}\frac{\varphi^2(x)}{|x|}\,dx\leq\|\varphi\|_{\mathcal D^{1/2,2}({{\mathbb R}}^N)}^2
  ,\quad\text{for all }\varphi\in \mathcal D^{1/2,2}({{\mathbb R}}^N).$$ Combining and we obtain that $$\label{eq:Sobolev-frazionario-bis}
S\big(N,\tfrac12\big)K^2 \, \|{\rm Tr\, }U\|_{L^{\frac{2N}{N-1}}({{\mathbb R}}^N)}^2\le
\|U\|_{\mathcal D^{1,2}({{\mathbb R}}^{N+1}_+)}^2 \qquad \text{for any }
U\in \mathcal D^{1,2}({{\mathbb R}}^{N+1}_+) .$$ Similarly, combining with , we infer $$\label{eq:Hardy-frazionario}
\Lambda K^2 \int_{{{\mathbb R}}^N} \frac{|{\rm
Tr\,}U|^2}{|x|} \, dx {\leqslant}\|U\|_{\mathcal D^{1,2}({{\mathbb R}}^{N+1}_+)}^2  \qquad \text{for any } U\in \mathcal
D^{1,2}({{\mathbb R}}^{N+1}_+) .$$ We recall the following lemmas from [@FF], which provide Sobolev and Hardy type trace inequalities 