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 Lévy 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 Itô'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.