Parabolic Littlewood-Paley inequality for φ(-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 $\phi(-\Delta)$, where $\phi$ is a Bernstein function. As an application, we construct an $L_p$-theory for the stochastic integro-differential equations of the type $du=(-\phi(-\Delta)u+f)dt +gdW_t$.