Norms of structured random matrices

For $m,n\in\mathbb{N}$ let $X=(X_{ij})_{i\leq m,j\leq n}$ be a random matrix, $A=(a_{ij})_{i\leq real deterministic and $X_A=(a_{ij}X_{ij})_{i\leq the corresponding structured matrix. We study expected operator norm of $X_A$ considered as between $\ell_p^n$ $\ell_q^m$ for $1\leq p,q \leq \infty$. prove optimal bounds up to logarithmic terms when underlying matrix $X$ has i.i.d. Gaussian entries, independent mean-zero bounded or $\psi_r$ ($r\in(0,2]$) entries. In certain cases, we determine precise order constants. Our results are expressed through sum norms Hadamard products $A\circ A$ $(A\circ A)^T$.