C_t  \sum_{I}\sum_{k=1}^{s}
\| N^k \psi\|_{W^{t-k-1/2}(b\Omega)}\\
& \le C_t \sum_{I}\sum_{k=1}^{s} \|N^k \psi\|_{t-k}\\
& \le C_t \|\psi\|_t ,\end{aligned}$$ where we use the assumption $t>s+1/2$ in order to able be to apply the trace theorem.

Finally notice that $\psi\llcorner\dbar\varrho=0$ in a neighborhood of $b\Omega$, $P^{(I)}_{s,k}$ becomes an operator of one degree lower, i.e., of order $s+k-1$. Repeating the argument above, we obtain that, for $t>s+1/2$ $$\|\K\psi\|_{t-1} \le C_t \|\psi\|_{t-1} 
.$$ This concludes the proof of the corollary.

[**Final Remarks.**]{} The results of Section 3 are obtained under a specific formulation of the Sobolev inner product. If we modify the formulation by choosing other positive coefficients $\gamma_\alpha$ in the definition of the inner product (\[Sobolev\]), results analogous to those presented here should still hold. It is also the case that the formulas that arise in these formulations of the norm are probably much less tractable.

The situation seems quite different if we take a generic equivalent norm. Consider, for instance, the weighted theory of the $\dbar$-Neumann problem, as developed by Kohn in [@Kohn]. Kohn showed that the regularity properties enjoyed by the canonical solution in the weighted case are in general much stronger than the ones enjoyed by the classical canonical solution (see also 