 prove that for $b\in C^{\infty}_0(\mathbb{R}^n)$, $\mathcal{M}_{\Omega,\,b}$ is completely continuous on $L^{p,\,\lambda}(\mathbb{R}^n,\,w)$.  Let $j_0\in\mathbb{Z}_-$, $b\in C^{\infty}_0(\mathbb{R}^n)$ with ${\rm supp}\, b\subset B(0,\,R)$