Reverse square function estimates for degenerate curves and its applications

Building on the classical work of Córdoba–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 $\{(\xi ,{\xi }^a):\mathrm{\mid }\xi \mathrm{\mid }\le 1\}$ in ${\mathbb{R}}^2$, for all exponents $a\in (0,\mathrm{\infty })\mathrm{\backslash }\{1\}$. As applications, we derive sharp $L^4$ Strichartz estimates on the one-dimensional torus for fractional Schrödinger equations and establish new local smoothing estimates in modulation spaces. In the latter application, orthogonal Strichartz-type estimates also play a crucial role.