A short proof of an identity related to Type IV superorthogonality

We provide a much shorter but even more powerful proof of an algebraic identity, which can be used to establish the direct and the converse inequality under Type IV superorthogonality. As an application, we obtain the optimal order of the formal constant in the direct inequality. This order turns out to be also sharp for Type III superorthogonality. When $p\geq 2$, our result recovers the optimal order of the constant in the Burkholder-Gundy inequality in martingale theory, and also recovers the currently best order of the constant in the reverse Littlewood-Paley inequality. We also introduce variants of Type IV superorthogonality, under which we prove the converse inequality.