well known properties of binomial coefficients. Assume the statement true for $p-1$ and all $m\le p-1$. Since $$F_k (p,m)=F_k (p-1,m)+F_k (p-1,m-1),$$ equality (\[claim\]) follows by induction and the equality in the case $m=p$. This finishes the proof of the ellipticity of (\[ellipticity\]).

Finally, if all the boundary data $g_{\ell}$ in problem (\[ellipticity\]) are identically $0$, then the only solution of the boundary value problem is the trivial one. In fact, if $u$ is such a solution, the identity (\[K-identity\]) with $u$ in place of $\K \psi $ implies that $u$ is orthogonal in the $W^s$ sense to all $\phi \in C^{\infty} (\Omega)$, hence $u=0$.

Finally, we have:

[Proof of Corollary 3.4]{} Clearly, $\K$ is well defined as composition of differential operators, restriction to the boundary, and the operator $S$ solution of the  in the previous Lemma.

Next, we use standard estimates for elliptic s, as in [@LiMa] Theorem 5.1, and Lemma \[ellipticity\]. Recall that $P^{(I)}_{s,k}$ is a differential operator of order $s+k$, containing $s$ at most derivatives in the normal direction. Then we see that for all $t>s+1/2$ $$\begin{aligned}
\| \K \psi\|_{t-1} 
& \le C_t \sum_I \| (\K \psi)_I\|_{t-1}  \\
& \le C_t \sum_{I}\sum_{k=1}^{s}
     \| P^{(I)}_{s,k} \psi\|_{W^{t-1-
(s+k-1)-1/2}(b\Omega)}\\
& \le 