For integers n≥ 2 and 1≤ p≤∞, let lambda_3(n,k,p,q) be the optimal constant in the inequality |u^(k)|_L^q(0,1)≤ lambda_3(n,k,p,q)|u^(n)|_L^p(0,1) for u∈mathring W_p^n(0,1). The Nazarov--Shcheglova conjecture asserts that lambda_3(n,1,p,1)=2 lambda_3(n,0,p,∞) for every n≥ 2 and 1≤ p≤∞. It further asserts that the corresponding extremal functions for these two inequalities coincide and are symmetric about the midpoint, namely satisfy u(x)=u(1-x) for x∈[0,1].

Nazarov–Shcheglova conjecture for sharp Sobolev inequalities

For integers $n\ge 2$ and $1\le p\le\infty$ , let $\lambda_3(n,k,p,q)$ be the optimal constant in the inequality $\|u^{(k)}\|_{L^q(0,1)}\le \lambda_3(n,k,p,q)\|u^{(n)}\|_{L^p(0,1)}$ for $u\in\mathring W_p^n(0,1)$ . The Nazarov--Shcheglova conjecture asserts that $\lambda_3(n,1,p,1)=2\lambda_3(n,0,p,\infty)$ for every $n\ge 2$ and $1\le p\le\infty$ . It further asserts that the corresponding extremal functions for these two inequalities coincide and are symmetric about the midpoint, namely satisfy $u(x)=u(1-x)$ for $x\in[0,1]$ .

References

Primary source

arXiv

Additional references

On the Nazarov--Shcheglova Conjecture for Sharp Sobolev Inequalities: The Case (n,p)=(3,2) — arXiv — Mohamed Jleli, Bessem Samet

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new paper settles the cubic square-integrability case, but the broader conjecture remains open.

The conjecture asserts that, for every $n\ge 2$ and $1\le p\le\infty$ , $\lambda_{3}(n,1,p,1)=2\lambda_{3}(n,0,p,\infty)$ , with matching symmetric extremizers.

Known results

The case $n=2$ is known for every $p$ , with $\lambda_{3}(2,1,p,1)=\frac14(p'+1)^{-1/p'}$ .

Cubic $L^{2}$ case reported on September 28, 2026

Mohamed Jleli and Bessem Samet report solving the case $(n,p)=(3,2)$ , including $\lambda_{3}(3,1,2,1)=\frac{1}{32\sqrt{5}}$ . This is progress on a specific parameter case, not a resolution for general $n$ and $p$ .

Current status (as of September 2026): the $n=2$ case is known and the $(n,p)=(3,2)$ case is claimed solved, while the conjecture for general $n$ and $p$ remains open.

Sources

No solutions have been posted yet.