On the strong unique continuation property for the Dirac operator

In [DO99,KY99], the strong unique continuation property from the origin is established for $H_{loc}^1$-solutions to the massless Dirac differential inequality $\mathrm{\mid }{D}_{n}u\mathrm{\mid }\le \frac{C}{\mathrm{\mid }x\mathrm{\mid }}\mathrm{\mid }u\mathrm{\mid }$, in dimension $n\ge 2$ and with $C<\frac{1}{2}$. We show that $\frac{1}{2}$ is the largest possibile constant in this result, providing an example in ${\mathbb{R}}^2$ of a (non-trivial) solution of the inequality. Also, we show properties of unique continuation from the origin for solutions to the inequality $\mathrm{\mid }{D}_{n}u\mathrm{\mid }\le \frac{C}{\mathrm{\mid }x{\mathrm{\mid }}^{\gamma }}\mathrm{\mid }u\mathrm{\mid }$, for $\gamma >1$, $C>0$. Finally, we establish the strong unique continuation property for the Dirac operator from the point at infinity.