Thanks, I have corrected this. I had in mind the fact that $\zeta(\bar z)=\overline{\zeta(z)}$.

Try the discrete dbar and its adjoint: both should have homogeneity $-1$ and their convolution should provide the fundamental solution of the Laplace operator.

@CPJ Thanks for your comment. In fact with $M_n=n^{ns}$, we get $(M_n/n^n)^{1/n}=n^{s-1}$ which is increasing for $s\ge 1$. This suggests that the composition algebra property holds for $s\ge 1$ but not for $s<1$.

@Piero D'Ancona You mean $s'<s$ since the Gevrey space $G^{(s)}$ with the notation above is increasing with $s$, e.g. analytic is $G^{(1)}$ is included in $G^{(2)}$ which contains compactly supported functions. On the other hand, I believe that the answer to the question is positive and is a matter of writing a precise Faà de Bruno formula.

There was a typo (now erased) with an unwanted ' in the last sentence.