Gagliardo-Nirenberg Inequality

The problem

Let $1\le q\le \infty$, and let $j$ and $m$ be non-negative integers such that $j<m$. Furthermore, let $1\le r\le \infty ,p\ge 1$ be real and $\theta \in [0,1]$ such that the following relations hold: $$\frac{1}{p}=j+\theta \left(\frac{1}{r}-m\right)+\frac{1-\theta }{q},\frac{j}{m}\le \theta <1.$$ Let $C(j,p,q,r,m)$ be the best constant such that $$\Vert D^{j}u{\Vert }_{L^p(\mathbb{R})}\le C(j,p,q,r,m)\Vert D^{m}u{\Vert }_{L^r(\mathbb{R})}^{\theta }\Vert u{\Vert }_{L^q(\mathbb{R})}^{1-\theta }$$ 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.

AlphaEvolve matched the known optimum or best construction.

What counts as progress

• A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
• A proof that a known construction is optimal, or a better bound on the other side.
• Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.