Reverse square function estimates for degenerate curves and their 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 $\delta$-neighborhood of the curve $\{(\xi ,|\xi {|}^a):|\xi |\le 1\}$ 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.

arXiv comment24 pages Corrected following referee report; he grammatical error in title fixed