An lp-boundedness of stochastic singular integral operators and its application to spdes

Abstract

In this article we introduce a stochastic counterpart of the Hörmander condtion on the kernel $K(r,t,x,y)$: there exists a pseudo-metric $\ddot{I}$ on $(0,\infty )\times R^d$ and a positive constant $C_0$ such that for $X=(t,x),Y=(s,y),Z=(r,z)\in (0,\infty )\times R^d$, $$\underset{X,Y}{\sup }{\int }_0^{\infty }{\left[{\int }_{\ddot{I}(X,Z)\ge C_0\ddot{I}(X,Y)}|K(r,t,z,x)-K(r,s,z,y)|dz\right]}^{2}dr<\infty .$$ We prove that the stochastic singular integral of the type $$\mathbb{T}g(t,x):={\int }_0^t{\int }_{R^d}K(t,s,x,y)g(s,y)dyd{W}_s$$ is a bounded operator on ${\mathbb{L}}_p=L_p(\hat{I}\text{c◯}\times (0,\infty );L_p(R^d))$ for any $p\ge 2$ if it is bounded when $p=2$ and stochastic Hörmander condition holds. Here $\hat{I}\text{c◯}$ is a probability space and $W_t$ is a Wiener process on $\hat{I}\text{c◯}$. Proving the $L_p$-boundedness of such integral operators is the key step in constructing an $L_p$-theory for linear stochastic partial differential equations (SPDEs in short). As a byproduct of our result on stochastic singular operators we obtain the maximal $L_p$-regularity result for a very wide class of SPDEs.