We obtain the boundedness in $L^p$ spaces for all $1<p<\infty$ of the so-called vertical Littlewood--Paley functions for

non-local Dirichlet forms in the metric measure space under some mild assumptions. For $1<p\le 2$, the pseudo-gradient is used to overcome the difficulty that chain rules are not valid for non-local operators, and then the Mosco convergence is employed to achieve the general case from the finite jumping kernel case, while for $2\le p<\infty$, the Burkholder--Gundy inequality is effectively applied.

The former method is analytic and the latter one is probabilistic. The results extend those ones for pure jump symmetric L\'evy processes

in Euclidean spaces. The talk is based on a joint work with Jian Wang (see arXiv:1704.02690v2).

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