$L^p-L^q$ estimates for convolutions with distribution kernels having singularities on the light cone

We study the convolution operator $T^z$ with the distribution kernel given by analytic continuation from the function $$ \widetilde{K}^z(y,s,t)= \left\{\begin{array}{ll} (t^2-s^2-|y|^2)_+^z/\Gamma(z+1)\quad &\mbox{if}\quad t>0\\ 0 \quad&\mbox{if} \quad t\le 0\end{array}\right\}, \quad Re(z)>-1 $$ where $(y,s,t)\in \mathbb R^{n-1}\times\mathbb R\times \mathbb R$. We obtain some improvement upon the previous known estimates for $T^z$. Then we extend the result of the cone multiplier of negative order on $\mathbb{R}^3$ \cite{lee1} to the case of general $\mathbb{R}^{n+1},\, n \ge 2$.