that give the initial conditions in (\[ODE\]). We then prove that the only bounded solution of (\[ODE\]) is in fact the trivial solution.

The boundary equations in (\[BVP2\]) arise from the identity (\[K-identity\]). By considering forms of the type $\phi_J
d\bar z^J$ we may reduce to the case of functions. We set $u=(\K\psi)_J$. Consider the top order terms in (\[K-identity\]), change coordinates, and freeze the coefficients. Write $\alpha=(k,\alpha')$ and notice that $\CP=\binom{s-1}{k}\CPp$. Then ${\partial}^\alpha =\po^k {\partial}^{\alpha'}$. Notice that the top order term in $T_{\alpha,|\alpha|-\ell}$ equals ${\partial}^{\alpha'}$, and that $T^*_{\alpha,|\alpha|-\ell}=(-1)^{|\alpha'|}{\partial}^{\alpha'}$. Then we have that $$\begin{aligned}
B_{s,\ell}
& = \sum_{|\alpha'|=0}^{s-1-\ell} \bn{|\alpha'|+\ell}{\ell} \CPp 
(-1)^{|\alpha'|} {\partial}^{2\alpha'} \po^{\ell+1} 
(-\Delta)^{s-1-\ell-|\alpha'|} \\
& = \sum_{j=0}^{s-1-\ell} \bn{j+\ell}{\ell}(-\Delta')^j
(-\Delta)^{s-1-\ell-j} \po^{\ell+1} ,\end{aligned}$$ where $\Delta'$ is the tangential Laplacian. Now write $\Delta=\po^2 +\Delta'$. We claim that the following identity holds true $$\label{combinatorics}
\sum_{j=0}^{s-1-k-\ell} \bn{\ell+j}{j}\bn{s-j-\ell-1}{k}
= \bn{s}{\ell+k+1}.$$ Assume the claim for now. Then, it turns out that $$B_{s,\ell} = \sum_{k=0}^{s-1-k} (-1)^k \bn{s}{\ell+k+1}
|\xi|^{2(s-1-\ell-k)}\po^{\ell+2k+1} .$$

Next, let $v=v_\xi$ be a bounded solution of (\[ODE\]) for $\xi\neq0$. Notice that $v=\bigl(\sum_{\ell=0}^{s-1}c_\ell x_0^\ell\bigr)e^{-|\xi|x_0}$. Let $f\in C^\infty_0 (\overline{{\Bbb R}^{2n-1}_+})$. Then for any $\xi\neq0$, by assumption and by integrating by parts we have $$\begin{aligned}
0 & = -\sum_{\ell=0}^{s-1} \po^\ell f (0,\xi) \overline{B_{s,\ell} 
v_\xi (0)} +\int_0^\infty f(x_0,\xi) 
\overline{(-\po^2 +|\xi|^2)^s 