Parabolic Littlewood-Paley inequality for $\varphi (-\mathrm{\Delta })$-type operators and applications to Stochastic integro-differential equations

In this paper we prove a parabolic version of the Littlewood-Paley inequality for the operators of the type $\varphi (-\mathrm{\Delta })$, where $\varphi$ is a Bernstein function. As an application, we construct an $L_p$-theory for the stochastic integro-differential equations of the type $du=(-\varphi (-\mathrm{\Delta })u+f)dt+gd{W}_t$.