Cotype 2 extrapolation on finite-dimensional linfinity

If a Banach space $Y$ has cotype 2 constant $C$, every bounded linear operator $T:{\ell }_{\infty }^N\to Y$ satisfies ${\pi }_2(T)\le 3^{1/4}C{\pi }_4(T)$ and ${\pi }_2(T)\le 4\sqrt{3}{C}^2\Vert T\Vert$, uniformly in $N$. The first bound follows from Pietsch factorization theorem and the fourth moment bound for Rademacher sums. For the second, 2-summing domination gives a weighted coordinate $L^2$ norm; clipping coordinates at a variable threshold yields ${\pi }_4(T)\le 2\sqrt{\Vert T\Vert {\pi }_2(T)}$, which absorbs the first bound.