On a family of one-dimensional oscillation inequalities

Abstract

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.