  Under simple hypotheses on the nonlinearity $f$, we consider the fractional
harmonic operator problem
\begin{equation}\label{abstr}\sqrt{-\Delta+|x|^2}\,u=f(x,u)\ \ \textrm{in }\
\mathbb{R}^N\end{equation} or, since we work in the extension setting
$\mathbb{R}^{N+1}_+$, $$\left\{\begin{aligned} -\Delta v +|x|^2v&=0,
&&\mbox{in} \ \mathbb{R}^{N+1}_+,\\ -\displaystyle\frac{\partial v}{\partial
x}(x,0)&=f(x,v(x,0)) &&\mbox{on} \ \mathbb{R}^{N}\cong\partial
\mathbb{R}^{N+1}_+.\end{aligned}\right.$$ Defining the space
$$\mathcal{H}(\mathbb{R}^{N+1}_+)=\left\{v\in H^1(\mathbb{R}^{N+1}_+):
\iint_{\mathbb{R}^{N+1}_+}\left[|\nabla v|^2+|x|^2v^2\right]dx
dy<\infty\right\},$$ we prove that the embedding
$$\mathcal{H}(\mathbb{R}^{N+1}_+)\hookrightarrow L^{q}(\mathbb{R}^N)$$ is
compact. We also obtain a Pohozaev-type identity for this problem, show that in
the case $f(x,u)=|u|^{p^*-2}u$ the problem has no non-trivial solution, compare
the extremal attached to this problem with the one of the space
$H^1(\mathbb{R}^{N+1}_+)$, prove that the solution $u$ of our problem belongs
to $L^p(\mathbb{R}^N)$ for all $p\in [2,\infty]$ and satisfy the polynomial
decay $|u(x)|\leq C/|x|$ for any $|x|>M$. Finally, we prove the existence of a
solution to a superlinear critical problem in the case
$f(x,u)=|u|^{2^*-2}u+\lambda |u|^{q-1}$, $1<q<2^*-1$.
