Submit progress

on Special Case of Gagliardo-Nirenberg Inequality (AlphaEvolve problem 16)

The statement

Let $2 < p < \infty$ . Let $C(p)$ denote the supremum of the quantities

Q(f) \coloneqq \frac{\|f\|_{L^p(\mathbb{R})}^{4p}}{\|f\|_{L^2(\mathbb{R})}^{2(p+2)} |f'|_{L^2(\mathbb{R})}^{2(p-2)}}

for all smooth rapidly decaying $f$ , not identically zero. Establish upper and lower bounds for $C(p)$ that are as strong as possible.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.