Fractional integration with singularity on the light-cone

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.