Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$\mathbf{L}^p$-boundedness of the Bochner-Riesz operator

Zipeng Wang

TL;DR

The paper proves ${L^p}$-boundedness of the Bochner-Riesz operator ${\bf S}^{\deltaup}$ for ${\rm Re}\,\deltaup\in(0,\tfrac{1}{2})$ within the range ${n-1\over 2n}\le{1\over p}\le{n+1\over 2n}$ by introducing a new approach based on cone multipliers of negative orders. It rewrites ${\left(1-|\xi|^2\right)^{\deltaup}_+}$ through an integral involving a negative-order cone multiplier ${\Lambda^\alphaup}$ and builds the analytic operator ${\bf I}^{\alphaup}$, with ${1\over 2}<{\rm Re}\alphaup<1$, to connect Bochner-Riesz summability with cone-restriction phenomena. A dual-stage dyadic decomposition in frequency and physical space paired via a Fefferman-Seeger-Sogge-Stein style scheme yields two families of analytic pieces ${\bf II}$ and ${\bf III}$, whose endpoint estimates (sharp and flat variants) are established and then interpolated to obtain the main ${L^p}$ bound. The results strengthen the link between Bochner-Riesz theory and cone multipliers, providing a robust framework for L^p estimates through endpoint control and complex interpolation. The techniques have potential implications for related restriction-type problems and other summability methods in harmonic analysis.

Abstract

In this paper, we give a new approach to the Bochner-Riesz summability. As a result, we show that the Bochner-Riesz operator $\mathbf{S}^δ, 0<\Reδ<{1\over 2}$ is bounded on $\mathbf{L}^p(\mathbb{R}^n)$ for ${n-1\over 2n}\leq {1\over p}\leq{n+1\over 2n}$.

$\mathbf{L}^p$-boundedness of the Bochner-Riesz operator

TL;DR

The paper proves -boundedness of the Bochner-Riesz operator for within the range by introducing a new approach based on cone multipliers of negative orders. It rewrites through an integral involving a negative-order cone multiplier and builds the analytic operator , with , to connect Bochner-Riesz summability with cone-restriction phenomena. A dual-stage dyadic decomposition in frequency and physical space paired via a Fefferman-Seeger-Sogge-Stein style scheme yields two families of analytic pieces and , whose endpoint estimates (sharp and flat variants) are established and then interpolated to obtain the main bound. The results strengthen the link between Bochner-Riesz theory and cone multipliers, providing a robust framework for L^p estimates through endpoint control and complex interpolation. The techniques have potential implications for related restriction-type problems and other summability methods in harmonic analysis.

Abstract

In this paper, we give a new approach to the Bochner-Riesz summability. As a result, we show that the Bochner-Riesz operator is bounded on for .
Paper Structure (16 sections, 2 theorems, 284 equations, 2 figures)

This paper contains 16 sections, 2 theorems, 284 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cone multiplier of negative orders
  3. A pairing formulation
  4. Theorem Two implies Theorem One
  5. End-point estimates regarding ${\bf II}^{\alphaup~\ell}_{j~m}$ and ${\bf III}^{\alphaup~\ell}_{j~m}$
  6. ${^\sharp}{\bf U}^{\alphaup~\betaup~\ell}_{j~m}$ and ${^\sharp}{\bf V}^{\alphaup~\betaup~\ell}_{j~m}$
  7. ${^\flat}{\bf U}^{\alphaup~\betaup~\ell}_{j~m}$ and ${^\flat}{\bf V}^{\alphaup~\betaup~\ell}_{j~m}$
  8. Proof of Theorem Two
  9. Proof of Proposition One
  10. Proof of Lemma One
  11. Size of ${^\sharp}\widehat{{\bf U}}^{\alphaup~\betaup~\ell}_{j~m}({\xiup})$ and ${^\sharp}\widehat{{\bf V}}^{\alphaup~\betaup~\ell}_{j~m}({\xiup})$
  12. On the ${\bf L}^1$-norm of ${^\flat}{\bf U}^{\alphaup~\betaup~\ell}_{j~m}$ and ${^\flat}{\bf V}^{\alphaup~\betaup~\ell}_{j~m}$
  13. Proof of Proposition Two
  14. Proof of Lemma Two
  15. Some estimates regarding Bessel functions
  16. ...and 1 more sections

Key Result

Lemma 5.1

Let ${1\over 10}<|{\xiup}|\leq 10$ and ${1\over 3}<|{\etaup}|\leq3$. Suppose ${\xiup}^\nu_j={\xiup}^\mu_j$ and ${\xiup}^{\overline{\mu}}_j\neq{\xiup}^\mu_j$ belonging to $\mathcal{Z}_\ell$. Either we have $\left|[{\bf L}_\mu^T{\xiup}-{\etaup}]_1\right|\approx2$ or $\left|[{\bf L}_\mu^T{\xiup}-{\etau

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (17)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Remark 5.1
  • Remark 5.2
  • Lemma 5.1
  • ...and 7 more