Capacities, Green function and Bergman functions

Using the logarithmic capacity, we give quantitative estimates of the Green function, as well as lower bounds of the Bergman kernel for bounded pseudoconvex domains in $\mathbb C^n$ and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies $K_\Omega(z)\gtrsim \delta_\Omega(z)^{-2}$ for any bounded pseudoconvex domain with $C^0-$boundary. An application to holomorphic motions is given.