Fractional integration with singularity on the light-cone

TL;DR

This paper investigates convolution operators with singularities on the light-cone, establishing key inequalities and regularity results for cone multipliers of negative order in harmonic analysis.

Contribution

It proves an open ${f L}^p o {f L}^q$ inequality and derives sharp regularity results for the cone multiplier problem of negative order.

Findings

Established ${f L}^p o {f L}^q$ norm inequality

Derived sharp regularity results for cone multipliers

Advanced understanding of singular convolution operators on the light-cone

Abstract

We study a family of convolution operators whose symbols and kernels have singularity on the light-cone in $\mathbb{R}^{n+1}$. First, we prove a desired ${\bf L}^p\to {\bf L}^q$ norm inequality which has been left open. Moreover, we obtain certain sharp regularity results for the cone multiplier problem of negative order.