Sharp $L^4$ Strichartz estimate for Hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$
Yangkendi Deng, Chenjie Fan, Zehua Zhao
TL;DR
This work establishes a sharp local $L^4$ Strichartz estimate for the hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$, removing derivative loss and matching the elliptic case on waveguides. The authors develop a kernel-decomposition framework together with semi-algebraic-set measure estimates to control the linear evolution $e^{it\square}$ and obtain a frequency-localized sharp bound, which then yields local and small-data global well-posedness for the cubic HNLS in $L^2(\mathbb{R}\times\mathbb{T})$. They also prove a sharp bilinear Strichartz estimate that captures precise dependence on frequency scales $N_1,N_2$ and the longitudinal period $\lambda$, illustrating optimal interaction bounds. The results advance the understanding of dispersion on waveguide geometries and provide tools applicable to related models such as the Davey–Stewartson system, while outlining directions for large-data theory and higher-dimensional generalizations.
Abstract
We prove the sharp $L^4$ Strichartz estimate without derivative loss for the hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$, \begin{equation} \|e^{it (\partial_{x_{1}}^2-\partial_{x_{2}}^2)} φ\|_{L^4_{t,x_{1},x_{2}}([0,1]\times \mathbb{R} \times \mathbb{T})}\lesssim \|φ\|_{L_{x_{1},x_{2}}^2(\mathbb{R} \times \mathbb{T})}, \end{equation} which serves as the hyperbolic analogue of the classical result of Takaoka-Tzvetkov \cite{takaoka20012d}. The proof is based on the combination of a robust kernel decomposition method with precise measure estimates for semi-algebraic sets. As an immediate application, we establish the global well-posedness for the cubic hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$ in the $L^2$-critical space with sufficiently small initial data.