General independent-entry matrix norm conjecture

Let $(X_{ij})_{i,j\leq n}$ be independent, mean-zero random variables satisfying condition

, and let $\alpha$ be the parameter appearing in that condition. Write $\|\cdot\|$ for the operator norm and $\|S\|_p=({\mathbb E}|S|^p)^{1/p}$. **General independent-entry matrix norm conjecture.** Then

\begin{aligned} {\mathbb E}|(X_{ij})| \sim_{\alpha} &\max_{i}\left(\sum_{j} {\mathbb E} X_{ij}^2\right)^{1/2}+\max_{j}\left(\sum_{i} {\mathbb E} X_{ij}^2\right)^{1/2} \ &+\max_{1\leq k\leq n}\min_{I\subset [n],|I|\leq k} \sup_{|s|2,|t|2\leq 1}\left|\sum{i,j\notin I}X{ij}s_it_j\right|_{\log (k+1)}. \end{aligned}

This is stated as a generalization of the Rademacher conjecture and would give a matching, up to constants depending on $\alpha$, upper bound for the corresponding lower estimate. The supplied \text does not reproduce condition

, so the precise scope of the conjecture should be checked in the paper.