Reverse square function estimates for degenerate curves and its applications

TL;DR

This paper proves reverse square function estimates for functions near degenerate curves in the plane and applies these results to obtain sharp Strichartz estimates and local smoothing results for fractional Schrödinger equations.

Contribution

It establishes $L^4$ reverse square function estimates for degenerate curves, extending classical results and enabling new applications in PDE analysis.

Findings

Proved $L^4$ reverse square function estimates for a range of degenerate curves.

Derived sharp $L^4$ Strichartz estimates on the torus for fractional Schrödinger equations.

Established new local smoothing estimates in modulation spaces.

Abstract

Building on the classical work of C\'{o}rdoba--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 $\{(\xi,\xi^a): |\xi|\leq 1\}$ 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 Schr\"{o}dinger equations and establish new local smoothing estimates in modulation spaces. In the latter application, orthogonal Strichartz-type estimates also play a crucial role.