Parabolic Littlewood–Paley inequality for ϕ(−Δ)ϕ(−Δ)-type operators and applications to stochastic integro-differential equations

In this paper we prove a parabolic version of the Littlewood–Paley inequality (1.4) for the operators of the type ϕ(−Δ)ϕ(−Δ), where ϕ   is a Bernstein function. As an application, we construct an LpLp-theory for the stochastic integro-differential equations of the type du=(−ϕ(−Δ)u+f)dt+gdWt.