中文摘要：

通过选择适当的L^p函数并应用连续分解方法，给出了低于临界阶的Bochner—Riesz算子在L^p空间有界的新的证明，同时得到了该算子和Lipschitz函数构成的高阶交换子L^p有界性的必要条件．

英文摘要：

In terms of continuous decomposition and choosing an appropriate L^p function, the author poses a new method to prove the necessary condition for L^p boundedness of Bochner-Riesz operators below the critical index. And a similar necessary condition for L^p-boundedness of higher order commutators generated by Bochner-Riesz operators below the critical index and Lipschitz functions is obtained by this method.