Reverse square function estimates for degenerate curves and its applications

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 $\hat{I}´$-neighborhood of the curve $\{(\hat{I}¾,\hat{I}{¾}^a):|\hat{I}¾|\le 1\}$ 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.

24 pages Some typos in the previous version were corrected