A note on small cap square function and decoupling estimates for the parabola

TL;DR

This paper establishes sharp small cap square function and decoupling estimates for the parabola, extending known results to new ranges of cap dimensions and enhancing understanding of harmonic analysis techniques.

Contribution

It provides new sharp estimates for small cap decoupling on the parabola for a broader range of cap sizes, complementing existing results.

Findings

Established sharp decoupling estimates for small caps on the parabola.

Extended the range of cap dimensions for which decoupling estimates are known.

Results are sharp up to polylogarithmic factors.

Abstract

In this paper, we prove small cap square function and decoupling estimates for the parabola, where the small caps are essentially axis-parallel rectangles of dimensions $\delta\times \delta^\beta$ for $0\leq \beta\leq 1$. Our estimates complement the known results for $1 \leq \beta \leq 2$ and are sharp up to polylogarithmic factors.