On pointwise convergence of cone multipliers

Abstract

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