On Lewy’s unsolvability phenomenon. (English) Zbl 1252.32013
Summary: We study a ramification of Lewy’s unsolvability phenomenon within the Teodorescu space \(B^1_{\mathbb R}(\Omega,\mathfrak X)\) (the domain of the minimal closed extension of the Lewy operator) with \(\mathfrak X\) a complex Fréchet space. We show that the Lewy equation \(\overline{(X,Y)}(u)=(\psi'\circ T,0)\) has no solution \(u:\Omega\to \mathfrak X\) of the Teodorescu class \(B^1\) defined on a neighbourhood of a point \((\xi+i\eta,\tau)\in\mathbb H_1^{}\), provided that \(\psi\in\mathcal D(\overline \partial_t)\subset C(\mathbb R,\mathfrak X)\) is not real analytic in \(\tau\).
MSC:
| 32A40 | Boundary behavior of holomorphic functions of several complex variables |
| 32W99 | Differential operators in several variables |
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
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