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A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

Shengwen Gan, Shukun Wu

Abstract

We prove some weighted $L^p\ell^p$-decoupling estimates when $p=2n/(n-1)$. As an application, we give a result beyond the real interpolation exponents for the maximal Bochner-Riesz operator in $\mathbb{R}^3$. We also make an improvement in the planar case.

A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

Abstract

We prove some weighted -decoupling estimates when . As an application, we give a result beyond the real interpolation exponents for the maximal Bochner-Riesz operator in . We also make an improvement in the planar case.
Paper Structure (8 sections, 14 theorems, 128 equations)

This paper contains 8 sections, 14 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. When
  4. When
  5. Application
  6. Preliminaries
  7. Two dimensions
  8. Three dimensions

Key Result

Theorem 1.1

Suppose $Y$ is a union of unit balls in $B_R$. Denote $\alpha_n=\frac{1}{2n(3n-1)}$ when $n\geq3$ and $\alpha_n=\frac{n-1}{4n^2}=\frac{1}{16}$ when $n=2$. Also denote $p_n=\frac{2n}{n-1}$. Then for any function $f$ such that ${\rm supp}\widehat{f}\subset N_{R^{-1}}(S)$, we have

Theorems & Definitions (25)

  • Theorem 1.1
  • Corollary 1.2
  • proof : Proof of Corollary \ref{['main2']} assuming Theorem \ref{['main']}
  • Conjecture 1.3
  • Conjecture 1.4: Tao-weak-type-BR
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Theorem 2.2: GIOW Theorem 4.2
  • Lemma 2.3
  • ...and 15 more