Submit progress

on Gagliardo-Nirenberg Inequality (AlphaEvolve problem 15)

The statement

Let $1 \leq q \leq \infty$ , and let $j$ and $m$ be non-negative integers such that $j < m$ . Furthermore, let $1 \leq r \leq \infty, p \geq 1$ be real and $\theta \in [0, 1]$ such that the following relations hold:

\frac{1}{p} = j + \theta \left( \frac{1}{r} - m \right) + \frac{1 - \theta}{q}, \quad \frac{j}{m} \leq \theta < 1.

Let $C(j,p,q,r,m)$ be the best constant such that

\|D^j u\|_{L^p(\mathbb{R})} \leq C(j,p,q,r,m) \|D^m u\|_{L^r(\mathbb{R})}^\theta \|u\|_{L^q(\mathbb{R})}^{1-\theta}

for all test functions $u$ , where $D$ denotes the derivative operator $\frac{d}{dx}$ . Establish upper and lower bounds for $C(j,p,q,r,m)$ 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.