Littlewood--Paley estimates for pure-jump Dirichlet forms

We employ the recent generalization of the Hardy--Stein identity to extend the previous Littlewood--Paley estimates to general pure-jump Dirichlet forms. The results generalize those for symmetric pure-jump L\'evy processes in Euclidean spaces. We also relax the assumptions for the Dirichlet form necessary for the estimates used in previous works. To overcome the difficulty that It\^o's formula is not applicable, we employ the theory of Revuz correspondence and additive functionals. Meanwhile, we present a few counterexamples demonstrating that some inequalities do not hold in the generality considered in this paper. In particular, we correct errors that appear in previous works.