BMO estimates for stochastic singular integral operators and its application to PDEs with L\'{e}vy noise

Abstract

In this paper, we consider the stochastic singular integral operators and obtain the BMO estimates. As an application, we consider the fractional Laplacian equation with additive noises \bess du_t(x)=\Delta^{\frac{\alpha}{2}}u_t(x)dt+\sum_{k=1}^\infty\int_{\mathbb{R}^m}g^k(t,x)z\tilde N_k(dz,dt),\ \ \ u_0=0,\ 0\leq t\leq T, \eess where $\Delta^{\frac{\alpha}{2}}=-(-\Delta)^{\frac{\alpha}{2}}$, and $\int_{\mathbb{R}^m}z\tilde N_k(t,dz)=:Y_t^k$ are independent $m$-dimensional pure jump L\'{e}vy processes with L\'{e}vy measure of $\nu^k$. Following the idea of \cite{Kim}, we obtain the $q$-th order BMO quasi-norm of the $\frac{\alpha}{q_0}$-order derivative of $u$ is controlled by the norm of $g$.