Parabolic Littlewood–Paley inequality for a class of time-dependent pseudo-differential operators of arbitrary order, and applications to high-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 that this inequality gives a fundamental estimate for an LpLp-theory of high-order stochastic partial differential equations.