A Note on the Norm of the Embedding of ${\text{PW}}^2$ into ${\text{PW}}^q$

We improve the currently best known bound for the norm of the embedding of $P{W}^2$ into $P{W}^q$ for all $q>2$. We show that our bound is asymptotically optimal as $q\to \mathrm{\infty }$. Our approach combines the recently established theorem of optimal Paley-Wiener concentration with a Hardy-Littlewood-Pólya majorization argument.