Reverse square function estimates for degenerate curves and its applications
Aleksandar Bulj, Kotaro Inami, Shobu Shiraki
TL;DR
The paper establishes $L^4$ reverse square function estimates for functions whose Fourier support lies near the degenerate curve $(\xi,\xi^a)$, valid for all $a>0$ with $a\neq 1$, and introduces a curved, scale-uniform localization framework to control overlaps. The key result bounds $\|F\|_{L^4}$ in terms of a square-function of $F_\omega$, with a sharp $\,\delta$-loss rate governed by $\rho(a,b)=\max\{0,1/b-1/a,1/b-1/2\}$. These estimates yield sharp $L^4$ Strichartz inequalities on the 1D torus for fractional Schrödinger equations and drive new local smoothing results in modulation spaces, complemented by orthogonal Strichartz-type bounds. The work also provides sharpness results for the constants and applications to local well-posedness of the cubic NLS, illustrating the impact of reverse-square-function technology on dispersive PDEs in degenerate geometric settings.
Abstract
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 $a\in(0,\infty)\backslash\{1\}$. 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.