  This paper is devoted to uniform versions of the Hanson-Wright inequality for
a random vector with independent centered $\alpha$-subexponential entries,
$0<\alpha\le 1$. Our method relies upon a novel decoupling inequality and a
comparison of weak and strong moments. As an application, we use the derived
inequality to prove the restricted isometry property of partial random
circulant matrices generated by standard $\alpha$-subexponential random
vectors, $0<\alpha\le 1$.
