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

Jongchon Kim, Liang Wang, Chun Keung Yeung

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 $δ\times δ^β$ for $0\leq β\leq 1$. Our estimates complement the known results for $1 \leq β\leq 2$ and are sharp up to polylogarithmic factors.

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

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 for . Our estimates complement the known results for and are sharp up to polylogarithmic factors.
Paper Structure (9 sections, 9 theorems, 74 equations)

This paper contains 9 sections, 9 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['sq func esti thm']}
  3. Broad-Narrow reduction
  4. Broad-Narrow analysis
  5. Proof of Theorem \ref{['sq func esti thm']}
  6. The proof of Theorem \ref{['s cap']} and Corollary \ref{['cor']}
  7. The proof of Theorem \ref{['s cap']}
  8. The proof of Corollary \ref{['cor']}
  9. Sharpness

Key Result

Theorem 1.1

For $0\leq \beta\leq1$ and $p\geq2$, we have the following small cap square function estimate: where $C_p$ is a constant depending only on $p$, and $c>0$ is a positive constant. The bound is sharp up to the $|\log \delta|^{c}$ factor.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Local $L^2$-orthogonality; see e.g. Gan24, Lemma 2
  • Lemma 2.4: Narrow part
  • proof
  • ...and 4 more