ON THE L-p-BOUNDEDNESS OF THE STOCHASTIC SINGULAR INTEGRAL OPERATORS AND ITS APPLICATION TO L-p-REGULARITY THEORY OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

In this article we introduce a stochastic counterpart of the Hormander condition and Calderon-Zygmund theorem. Let W-t be a Wiener process in a probability space Omega and let K(omega, r, t, x, y) be a random kernel which is allowed to be stochastically singular in a domain O subset of R-d in the sense that E vertical bar integral(t)(0) integral vertical bar(x-y vertical bar<epsilon) vertical bar K(omega, s, t, y, x)vertical bar dydW(s)vertical bar(p) =infinity for all t, p, epsilon > 0, x is an element of O. We prove that the stochastic integral operator of the type (0.1) Tg(t, x) := integral(t)(0) integral(O) K(omega, s, t, y, x)g(s, y)dydW(s) is bounded on L-p = L-p (Omega x (0,infinity); L-p(O)) for all p is an element of [2,infinity) if it is bounded on L-2 and the following (which we call stochastic H<spacing diaeresis>ormander condition) holds: there exists a quasi-metric rho on (0,infinity) x O and a positive constant C-0 such that for X = (t, x), Y = (s, y), Z = (r, z) is an element of (0,infinity) x O, sup(omega is an element of Omega,X,Y) integral(infinity)(0) [integral(rho(X,Z)>= C0 rho(X,Y)) vertical bar K(r, t, z, x) - K(r, s, z, y)vertical bar dz](2) dr < infinity. Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp L-p-regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.