Parabolic Littlewood-Paley inequality for a class of time-dependent operators of arbitrary order, and applications to higher order stochastic PDE

In this paper we prove a parabolic version of the Littlewood-Paley inequality for a class of time-dependent local and non-local operators of arbitrary order, and as an application we show this inequality gives a fundamental estimate for the $L_p$-theory of the stochastic partial differential equations.