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

Special Case of Gagliardo-Nirenberg Inequality (AlphaEvolve problem 16). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 26D10 · Inequalities involving derivatives and differential and integral operatorsMSC 46E35 · Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
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Problem statement

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

Q(f)≔∥f∥Lp(R)4p∥f∥L2(R)2(p+2)∣f′∣L2(R)2(p−2)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 ff, not identically zero. Establish upper and lower bounds for C(p)C(p) that are as strong as possible.

Lean 4

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Background

Repository result category: matched optimal — AlphaEvolve matched the known or conjectured optimum. See Section 6.7 in "Mathematical Exploration and Discovery at Scale" paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

The repository page records no bounds for this problem.

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — Special Case of Gagliardo-Nirenberg Inequality (AlphaEvolve problem 16) linkApache-2.0 (repository license); problem pages reproduced with attribution.

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