There are many possible variants of our ÒProperty HÓ which will end up having the same effect. In particular one can avoid the $L^{p}$ theory. In the context we work in, we have a first derivative bound as in Prop. 2.1 (1), and it is easy to show using this that the $L^{2}$ norm of $\tau$ controls $|\tau(\chi(u_{*}))|$.