On almost everywhere convergence of Bochner-Riesz means in higher dimensions

Abstract:

In ${{\mathbf {R}}^n}$ define $({T_{\lambda ,r}}f)(\xi ) = \hat f(\xi )(1 - \left | {{r^{ - 1}}{\xi ^2}} \right |)_ + ^\lambda$. If $n \geq 3$, $\lambda > \tfrac {1}{2}(n - 1)/(n + 1)$ and $2 \leq p < 2n/(n - 1 - 2\lambda )$, then ${\lim _{r \to \infty }}{T_{\lambda ,r}}f(x) = f(x)$ a.e. for all $f \in {L^p}({{\mathbf {R}}^n})$.