In a recent paper of Green and Sanders entitled "Monochromatic sums and products" on page 11 of the Arxiv version there is the following.
We have a function say $f_1$ for which $\|f_1\|_2 \leq 1$ is known. However, nothing is assumed on $\|f_1\|_\infty$. They define another function $g_1(t) = f_1(t) \exp_p(at^2 +bt)$. For $g_1$ it is clear that $\|g_1\|_2 \leq 1$ but in this paper it is implied that $\|g_1\|_\infty = O(1)$.
More generally, if a function has its $L^2$ norm bounded by, say $1$, why is it that multiplying it by a quadratic exponential ensures its infinity norm is bounded by a constant?
I realise my question is bordering on the naive but I am a bit rusty here.