Reverse square function estimates for degenerate curves and their 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 $δ$-neighborhood of the curve $\{(ξ,|ξ|^a): |ξ|\leq 1\}$ in $\mathbb{R}^2$, for all exponents $a\in(0,\infty