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

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 ${\mathrm{\Delta }}^{\frac{\alpha }{2}}=-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}$, 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 $g$.