We intend to discuss global pointwise estimates for the kernel associated with the resolvent $(I - T)^{-1}$ of the integral operator $T f(x) = int K(x,y) f(x),dx$ on $L^2(Omega)$ for strict contractions under the only assumption that $d(x, y) = 1/ K(x, y)$ is a quasimetric. As an application, we give bilateral bounds for Green's function of the fractional Schrödinger operator $(-Delta)^{alpha/2} - q$ with an arbitrary nonnegative potential $q$ on $R^n$ for $0 < alpha < n$, or a bounded Lipschitz domain for $0 < alpha le 2$. This yields explicit bounds for the conditional gauge associated with Brownian motion or $alpha$ -stable Lévy processes. Other applications include necessary and sufficent conditions for the existence of weak solutions to the Dirichlet problem for the nonlinear equation $-Delta u = |nabla u|^2 +q$ in terms of exponential integrability of the balayage of $domega = dist(., partial Omega) q dsigma$ on $partial Omega$. A sharp sufficient condition is that $omega$ is a Carleson measure and $|T| < 1$. This talk is based on joint work with Michael Frazier and Fedor Nazarov.