On Lewy's unsolvability phenomenon

We study a ramification of Lewy’s unsolvability phenomenon within the Teodorescu space $B^1_{\mathbb R} (\Omega, \mathcal{X})$ (the domain of the minimal closed extension of the Lewy operator) into a complex Fréchet space. We show that the Lewy equation $\overline{(X,Y)} (u) = (\psi^\prime \circ \mathcal{T}, 0)$ has no solution $u: \Omega \to \mathcal{X}$ of Teodorescu class $B^1$ , defined on a neighbourhood of a point $(\xi + i \eta, \tau) \in \mathbb{H}_1$ provided that $\psi \in \mathcal{D}(\overline{\partial}_t) \subset C(\mathbb{R}, \mathcal{X})$ is not real analytic at $\tau$.