SOME SHARP BOUNDS FOR THE CONE MULTIPLIER OF NEGATIVE ORDER IN ${\mathbb R}^3$

Abstract

This paper considers the cone multiplier operator which is defined by $\[\widehat{S^\mu f}(\xi,\tau)=m_\mu(\xi,\tau)\widehat f(\xi,\tau)$, $\qquad (\xi,\tau)\in \mathbb R^2\times \mathbb R\]$ where $m_\mu(\xi,\tau)=\phi(\tau)(1-|\xi|^2/\tau^2)_+^\mu/\Gamma(\mu+1)$ and $\phi\in C_0^\infty(1,2)$. For $-3/2<\mu<-3/14$, sharp $L^p-L^q$ estimates and endpoint estimates for $S^{\mu}$ are obtained.