Bochner-Riesz means on a conical singular manifold

TL;DR

This paper establishes a precise $L^p$-boundedness criterion for Bochner-Riesz multipliers on conical singular manifolds, extending to sectors with boundary conditions and resolving the critical exponent problem in these geometries.

Contribution

It provides a sharp $L^p$ boundedness criterion for Bochner-Riesz operators on conical and sector domains, including boundary conditions, which was previously unresolved.

Findings

Boundedness holds for $oxed{ ext{all } p eq 2}$ when $oxed{ ext{the order } oldsymbol{oldsymbol{ extdelta} > oldsymbol{oldsymbol{ extdelta}_c(p,2)}}$.

The critical exponent $oldsymbol{oldsymbol{ extdelta}_c(p,2)}$ is explicitly characterized as $oldsymbol{ ext{max}igrace{0, 2|1/2 - 1/p| - 1/2}ig}$.

Results apply to flat cones and sectors with boundary conditions, resolving the critical exponent problem in these settings.

Abstract

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