A poisson equation associated with an integral kernel operator

Suppose the kernel function $\kappa$ belongs to $S(R^2)$ and is symmetric such that $ < \otimes x, \kappa >\geq 0$ for all $x \in S'(R)$. Let A be the class of functions f such that the function f is measurable on $S'(R)$ with $\int_{S'(R)}$\mid$f((I + tK)^{\frac{1}{2}}x$\mid$^2d\mu(x) < M$ for some $M > 0$ and for all t > 0, where K is the integral operator with kernel function $\kappa$. We show that the \lambda$-potential $G_Kf$ of f is a weak solution of $(\lambda I - \frac{1}{2} \tilde{\Xi}_{0,2}(\kappa))_u = f$.