关于一类一维振荡不等式
On a family of one-dimensional oscillation inequalities
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中文总结 AI 辅助
该研究针对一维环面上的非零连续均值为零函数，证明了一类最优振荡不等式，解决了S. Steinerberger提出的公开问题，基于类Gagliardo-Nirenberg估计等完成证明，并推导了相关应用结果。
中文摘要 AI 辅助
设φ是一维环面上的非零连续均值为零函数，记N_φ为φ变号的次数。我们证明了如下形式的最优振荡不等式族：\n\begin{equation*} N_\varphi\\|\varphi\\|_{\dot W^{-1,s}} \gtrsim_{p,s} \frac{\\|\varphi\\|_1^{1+p'/s}}{\\|\varphi\\|_p^{p'/s}} \\, \\, \text{ 和 } \\, \\, (N_\varphi)^\alpha\\|\phi\\|_{\dot W^{-1,s}} \gtrsim_{p,q,r,s,\alpha} \frac{\\|\varphi\\|_p\\|\varphi\\|_q}{\\|\varphi\\|_r}. \end{equation*}\n这一结果解决了S. Steinerberger提出的公开问题，且强化了原始估计。证明不依赖最优传输，基于类Gagliardo-Nirenberg估计以及负Sobolev半范数的商空间刻画。我们还讨论了该结果在若干振荡问题中的应用，作为应用，推导了与傅里叶投影、不确定性原理以及Sturm-Hurwitz定理相关的若干振荡估计。
英文摘要
Let $φ$ be a nonzero continuous mean-zero function on the one-dimensional torus and let $N_φ$ be the number of times that $φ$ changes signs. We prove the sharp family of oscillation inequalities of the types \begin{equation*} N_φ\|φ\|_{\dot W^{-1,s}} \gtrsim_{p,s} \frac{\|φ\|_1^{1+p'/s}}{\|φ\|_p^{p'/s}} \, \, \text{ and } \, \, (N_φ)^α\|ϕ\|_{\dot W^{-1,s}} \gtrsim_{p,q,r,s,α} \frac{\|φ\|_p\|φ\|_q}{\|φ\|_r}. \end{equation*} This resolves an open problem posed by S. Steinerberger and strengthens the original estimate. The proof is independent of optimal transport and is based on a Gagliardo-Nirenberg-type estimate as well as a quotient-space characterization of the negative Sobolev seminorm. As applications, we derive several oscillation estimates related to Fourier projection, the uncertainty principle, and the Sturm-Hurwitz theorem.
