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arXiv · 2608.04639

On a family of one-dimensional oscillation inequalities

Fushuai Jiang
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Abstract

Let $\varphi$ be a nonzero continuous mean-zero function on the one-dimensional torus and let $N_\varphi$ be the number of times that $\varphi$ changes signs. We prove the sharp family of oscillation inequalities of the types \begin{equation*} N_\varphi\|\varphi\|_{\dot W^{-1,s}} \gtrsim_{p,s} \frac{\|\varphi\|_1^{1+p'/s}}{\|\varphi\|_p^{p'/s}} \, \, \text{ and } \, \, (N_\varphi)^\alpha\|\phi\|_{\dot W^{-1,s}} \gtrsim_{p,q,r,s,\alpha} \frac{\|\varphi\|_p\|\varphi\|_q}{\|\varphi\|_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.

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Fushuai Jiang. 2026-08-05. On a family of one-dimensional oscillation inequalities. https://arxiv.org/abs/2608.04639

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