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Is the product of sub-Gaussian polynomials in $\mathbb{R}[x]/(x^n-1)$ sub-Gaussian?

I had known that Or (in general) via some stronger Orlicz-norm bound on $f(x)g(x)$ than the above $\psi_1$ bound? was plausibly related to the Bennett-Orlicz norm (or perhaps Bernstein-Orlicz norm), …
Mark Schultz-Wu's user avatar