A family of convolution operators, part two
Zipeng Wang
TL;DR
The paper studies a family of convolution operators with Fourier multipliers defined from distributions supported on the light cone. It introduces $\mathbf{I}^{\alphaup}$ via $\Lambda^{\alphaup}$ and reformulates it through a multiplier $\widehat{\mathbf{P}}^{\alphaup}(\xi)$ tied to $\Omega^{\alphaup}$ and Bessel-function representations, proving $L^p$ bounds for ${1\over 2}<\Re\alphaup<1$ in the range ${n-1\over 2n}\le {1\over p}\le {n+1\over 2n}$. A decomposition into restricted ${\bf L}^p$-multipliers $({\bf m}^{\alphaup}_{+},{\bf m}^{\alphaup}_{-})$ and their combinations yields Theorem 1 on $L^p$-boundedness and Theorem 2 extending to Bochner-Riesz type operators ${\bf S}^{\deltaup}_{\psiup}$ for $0<\Re\deltaup<\tfrac{1}{2}$, linking light-cone analysis to summability. This provides a new analytic framework for Bochner-Riesz summability, with implications for the ball multiplier problem via a multiplier-based approach.
Abstract
We study a family of convolution operators. Their regarding Fourier multipliers are defined in terms of distributions having singularity on the light-cone in $\mathbb{R}^{n+1}$. As a result, we give a new approach to the Bochner-Riesz summability.