Parabolic Littlewood-Paley inequality for $\ddot{I}•(-\hat{I}”)$-type operators and applications to Stochastic integro-differential equations

Abstract

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