[For the entire collection see Zbl 0704.00024.]
The author proves the following theorem. Let \(\vec D\) be the Dirac operator in \({\mathbb{R}}^ 3\), \(\Omega \subset {\mathbb{R}}^ 3\) be an open, connected subset and v: \(\Omega\to {\mathbb{R}}\) a function in \(L^{3,5}_{loc}(\Omega)\). If \(\psi \in H^ 1_{loc}(\Omega)\), \(| {\vec \alpha}\vec D\psi (x)| \leq v(x) | \psi (x)|\) (a.e. in \(\Omega\)) and \(\psi (x)=0\) on an open, non-empty subset, then \(\psi\equiv 0.\)
The proof is based on the usual Carleman method, i.e. on the proof of inequalities of the type \(\| e^{\tau \phi}f\|_{L^ q(U)}\leq c \| e^{\tau \phi}{\vec \alpha}\vec Df\|_{L^ p(U)}\), where \(U=\{x\in {\mathbb{R}}^ 3|\) \(0<a<| x| <b<\infty \}\) (for some small b), p,q\(\geq 1\), \(f\in C_ 0^{\infty}(U)\), \(\tau\) is a real parameter and the constant c is independent of \(\tau\) and f.