发表机构
King Saud University（沙特国王大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Nazarov-Shcheglova猜想在(n,p)=(3,2)情形下成立，得到最优常数λ_3(3,1,2,1)=1/(32√5)，并刻画了所有极值函数。
AI 中文摘要
对于整数$n>k\geq0$和$1\leq p,q\leq\infty$，设$\lambda_3(n,k,p,q)$表示一维Sobolev不等式\\[ \\|u^{(k)}\\|_{L^q(0,1)} \leq \lambda_3(n,k,p,q) \\|u^{(n)}\\|_{L^p(0,1)}, \qquad u\in\mathring W_p^n(0,1) \\]中的最优常数。Nazarov和Shcheglova猜想\\[ \lambda_3(n,1,p,1) = 2\lambda_3(n,0,p,\infty), \qquad n\geq2,\quad 1\leq p\leq\infty, \\]并且相应的极值函数重合且关于区间中点对称。我们证明了该猜想在$(n,p)=(3,2)$时成立，并特别得到\\[ \lambda_3(3,1,2,1) = \frac{1}{32\sqrt5}。\\]证明基于将其归约为一个带有附加矩约束的算子范数问题。在重新标度到$(-1,1)$后，该约束导致与二次多项式正交。我们通过相应的正交投影扩展二阶导数的逆，并通过奇偶分解分析伴随算子。奇分量由一个正Gram核控制，而偶分量通过弱$L^2$估计处理。等式情形给出了所有极值函数的刻画。
英文摘要
For integers $n>k\geq0$ and $1\leq p,q\leq\infty$, let $λ_3(n,k,p,q)$ denote the optimal constant in the one-dimensional Sobolev inequality \[ \|u^{(k)}\|_{L^q(0,1)} \leq λ_3(n,k,p,q) \|u^{(n)}\|_{L^p(0,1)}, \qquad u\in\mathring W_p^n(0,1). \] Nazarov and Shcheglova \cite{NazarovShcheglova} conjectured that \[ λ_3(n,1,p,1) = 2λ_3(n,0,p,\infty), \qquad n\geq2,\quad 1\leq p\leq\infty, \] and that the corresponding extremal functions coincide and are symmetric about the midpoint of the interval. We prove this conjecture for $(n,p)=(3,2)$ and, in particular, obtain \[ λ_3(3,1,2,1) = \frac{1}{32\sqrt5}. \] The proof is based on a reduction to an operator norm problem with an additional moment constraint. After rescaling to $(-1,1)$, this constraint leads to orthogonality with respect to quadratic polynomials. We extend the inverse of the second derivative through the corresponding orthogonal projection and analyze the adjoint operator by an odd--even decomposition. The odd component is controlled by a positive Gram kernel, while the even component is treated by a weak-$L^2$ estimate. The equality cases yield the characterization of all extremal functions.
