On pointwise convergence of cone multipliers

Abstract

For $p\ge 2$, and $\lambda >\max \{n|\frac{1}{p}-\frac{1}{2}|-\frac{1}{2},0\}$, we prove the pointwise convergence of cone multipliers, i.e. $$\underset{t\to \infty }{\lim }{T}_t^{\lambda }(f)\to f\text{ a.e.},$$ where $f\in L^p({\mathbb{R}}^n)$ satisfies $supp\hat{f}\subset \{\xi \in {\mathbb{R}}^n:1<|{\xi }_n|<2\}$. Our main tools are weighted estimates for maximal cone operators, which are consequences of trace inequalities for cones.