How about calculating $\sqrt{B^*B}=\vert B\vert$?

@Denis Serre: I do not understand your Borel-type argument for the smoothness of $w^0$ at the origin. The size of $w_k$ on the sphere could be anything and your argument ("differs from…") would provide smoothness for $\sum_{k\ge 0}\phi(kx)a_k x^k$ for any sequence. Given a sequence $(a_k)_{k\ge 0}$, it is possible to choose a sequence $(\mu_k)_{k\ge 0}$ (depending heavily on the $a_k$) such that $\sum_{k\ge 0}\phi(\mu_k x)a_k x^k$ is smooth (i.e. $C^\infty$).

Thanks for this very nice counterexample. The vanishing of the mean (i.e. $\hat \nu(0)=0$) that I required in the question above is certainly necessary, but as you have just pointed out is not sufficient. All moments should vanish, this is an elegant addendum to Poincaré Lemma.

I should have made my answer more precise, since it deals indeed with real analyticity (I have modified my answer). Of course, ellipticity alone does not imply the sought property globally in $\mathbb C^n$: take for instance for $n=1$, $\frac{\partial}{\partial z}$ and the elliptic equation $\frac{\partial u}{\partial z}=0$, which has no (non-trivial) holomorphic function as a solution.