A BMO estimate for stochastic singular integral operators and its application to SPDEs

In this paper, we investigate kernel conditions on K(t,s,x)K(t,s,x) so that the stochastic singular integral operator ∫0tK(t,s,⋅)⁎g(s,⋅)(x)dws has a bounded mean oscillation. As an application, we prove that for the solution u of the stochastic heat equationequation(0.1)dut(x)=aij(t)uxixjdt+gtk(x)dwtk,u0=0,t≤T, the q-th order BMO quasi-norm of the derivatives of u   is controlled by ‖g‖L∞‖g‖L∞.