Fractional integration with singularity 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 ${\mathbf{L}}^p\to {\mathbf{L}}^q$ norm inequality which has been left open. Moreover, we obtain certain sharp regularity results for the cone multiplier problem of negative order.

A gap is found in the proof given in section 6 which cannot be filled. I believe that the correct proof requires a new framework