Why the raw root needs a collar

The smooth advanced function (eq:beta) may extend to negative times. In particular, it need not vanish at $t=0$. One cannot generally multiply it by a smooth cutoff supported in $t\ge 0$ which is identically one at every positive time.

Choose $\delta >0$ and $\vartheta \in {C_c}^{\infty }((-\delta ,T+\delta ))$ with $\vartheta =1$ on a neighborhood of $[0,T]$. Set