Sharp $L^4$ Strichartz estimate for Hyperbolic Schr\"odinger equation on $\mathbb{R}\times \mathbb{T}$

We prove the sharp $L^4$ Strichartz estimate without derivative loss for the hyperbolic Schr\"odinger equation on $\mathbb{R}\times\mathbb{T}$, \begin{equation} \|e^{it (\partial_{x_{1}}^2-\partial_{x_{2}}^2)} \phi\|_{L^4_{t,x_{1},x_{2}}([0,1]\times \mathbb{R} \times \mathbb{T})}\lesssim \|\phi\|_{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 Schr\"odinger equation on $\mathbb{R}\times\mathbb{T}$ in the $L^2$-critical space with sufficiently small initial data.