Iterated Cauchy-Schwarz bound for a Volterra operator ($\Vert {\tau }^{n}f(t)\Vert \le \frac{C^n{t}^n}{\sqrt{(2n-1)!!}}{\sup }_{s\le t}\Vert f(s)\Vert$)

Starting from $\Vert \tau g(t){\Vert }^2\le C^{2}t{\int }_0^t\Vert g(s){\Vert }^{2}ds$, induction integrates $s^{2n-2}$ at the $n$th step. Its denominator accumulates the odd integers $1,3,\dots ,2n-1$. The factorial bound for a Volterra iterate is stronger because $(n!{)}^2\ge (2n-1)!!$.