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Bochner-Riesz means on a conical singular manifold

Qiuye Jia, Junyong Zhang, Jiqiang Zheng

TL;DR

This work determines the sharp L^p-boundedness threshold for Bochner-Riesz means on two-dimensional flat cones, showing that S_λ^δ(Δ_X) is bounded on L^p(X) for p ≠ 2 precisely when δ > δ_c(p,2) with δ_c(p,2) = max{0, 2|1/2 − 1/p| − 1/2}. The authors construct an explicit spectral-measure kernel for Δ_X, decompose the Bochner-Riesz kernel into geometric and diffractive components, and apply Stein–Hörmander oscillatory integral theory together with cone-specific Young-type estimates to handle diffraction at the cone vertex. A key reduction to σ > 1 via averaging over angular orbits enables the analysis to proceed uniformly, and the method extends to infinite sectors with Dirichlet/Neumann boundaries, resolving the wedge critical-exponent problem. The results provide a sharp, geometry-aware BR theory in the conical setting and lay groundwork for polygonal-domain problems by clarifying diffraction and long-time dynamics in two dimensions.

Abstract

We prove a sharp $L^p$-boundedness criterion for Bochner-Riesz multipliers on flat cones $X = (0,\infty) \times \mathbb{S}_σ^1$. The operator $S_λ^δ(Δ_X)$ is bounded on $L^p(X)$ for $1 \leq p \leq \infty$, $p \neq 2$, if and only if $δ> δ_c(p,2) = \max\left\{ 0, 2\left| 1/2 - 1/p \right| - 1/2 \right\}$. This result is also applicable to the infinite sector domain with Dirichlet or Neumann boundary, resolving the critical exponent problem in this wedge setting.

Bochner-Riesz means on a conical singular manifold

TL;DR

This work determines the sharp L^p-boundedness threshold for Bochner-Riesz means on two-dimensional flat cones, showing that S_λ^δ(Δ_X) is bounded on L^p(X) for p ≠ 2 precisely when δ > δ_c(p,2) with δ_c(p,2) = max{0, 2|1/2 − 1/p| − 1/2}. The authors construct an explicit spectral-measure kernel for Δ_X, decompose the Bochner-Riesz kernel into geometric and diffractive components, and apply Stein–Hörmander oscillatory integral theory together with cone-specific Young-type estimates to handle diffraction at the cone vertex. A key reduction to σ > 1 via averaging over angular orbits enables the analysis to proceed uniformly, and the method extends to infinite sectors with Dirichlet/Neumann boundaries, resolving the wedge critical-exponent problem. The results provide a sharp, geometry-aware BR theory in the conical setting and lay groundwork for polygonal-domain problems by clarifying diffraction and long-time dynamics in two dimensions.

Abstract

We prove a sharp -boundedness criterion for Bochner-Riesz multipliers on flat cones . The operator is bounded on for , , if and only if . This result is also applicable to the infinite sector domain with Dirichlet or Neumann boundary, resolving the critical exponent problem in this wedge setting.

Paper Structure

This paper contains 13 sections, 19 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. The spectral measure and Bochner-Riesz mean
  3. The kernel of resolvent and spectral measure
  4. The kernel of Bochner-Riesz means
  5. Reduction
  6. Proof of Theorem \ref{['thm:main']}
  7. The contribution of $T_{G_e}$ and $T_{D_e}$
  8. The contribution of $T_{G_m^\pm}$
  9. The estimate of $T_{D_m}^\pm$
  10. Proof of Proposition \ref{['prop:keyTD1+k']}
  11. Appendix
  12. Basic harmonic analysis on cones
  13. Some basic lemmas

Key Result

Theorem 1.2

Let $1\leq p\leq+\infty$ and $\delta>\delta_c(p,2)$ be given by equ:BRconj, which means Then, there holds where the constant $C$ is independent of $\lambda>0$.

Theorems & Definitions (39)

  • Conjecture 1.1: Bochner-Riesz conjecture
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1: Spectral measure kernel
  • Remark 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6
  • ...and 29 more