Reverse square function estimates for degenerate curves and its applications

Building on the classical work of C\'{o}rdoba--Fefferman and the recent work of Schippa, we establish $L^4$ reverse square function estimates for functions whose Fourier support is contained in a $\delta$-neighborhood of the curve $\{(\xi,\xi^a): |\xi|\leq 1\}$ in $\mathbb{R}^2$, for all exponents $a\in(0,\infty)\backslash\{1\}$. As applications, we derive sharp $L^4$ Strichartz estimates on the one-dimensional torus for fractional Schr\"{o}dinger equations and establish new local smoothing estimates in modulation spaces. In the latter application, orthogonal Strichartz-type estimates also play a crucial role.