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 $|{D}_n u | \leq \frac{C}{|x|}|u|$, in dimension $n\geq 2$ and with $C<\frac12$. We show that $\frac12$ 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 $|D_n u | \leq \frac{C}{ |x|^{\gamma}}|u|$, for $\gamma>1$, $C>0$. Finally, we establish the strong unique continuation property for the Dirac operator from the point at infinity.