Iterated Cauchy-Schwarz bound for a Volterra operator
= Iterated Cauchy-Schwarz bound for a Volterra operator
{title2=$\|\tau^nf(t)\|\leq\frac{C^nt^n}{\sqrt{(2n-1)!!}}\sup_{s\leq t}\|f(s)\|$}
Starting from $\|\tau g(t)\|^2\leq C^2t\int_0^t\|g(s)\|^2ds$, induction integrates $s^{2n-2}$ at the $n$th step. Its denominator accumulates the odd integers $1,3,\ldots,2n-1$. The <factorial bound for a Volterra iterate> is stronger because $(n!)^2\geq(2n-1)!!$.