We obtain the boundedness in L^p spaces for all 1 < p < ∞ of the so-called vertical Littlewood-Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For 1 < p ⩽ 2, the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the uniformly bounded jumping kernel case to the general case, while for 2 ⩽ p ⩽ ∞, the Burkholder-Davis-Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those for pure jump symmetric Lévy processes in Euclidean spaces.

在某些温和的假设下，对于度量度量空间中非局部Dirichlet形式的所谓垂直Littlewood-Paley函数的所有1 <p <∞，我们获得L ^p空间中的有界性。对于1 <p⩽2，引入伪梯度以克服链规则不适用于非局部算子的难题，然后使用Mosco收敛为从均匀有界跳核案例到一般情况铺平道路。在2 while p∞的情况下，有效地应用了Burkholder-Davis-Gundy不等式。前一种方法是解析性的，后一种方法是概率性的。结果扩展了欧氏空间中纯跳跃对称Lévy过程的结果。