  We establish a uniform Sobolev inequality for K\"ahler metrics, which only
require an entropy bound and no lower bound on the Ricci curvature. We further
extend our Sobolev inequality to singular K\"ahler metrics on K\"ahler spaces
with normal singularities. This allows us to build a general theory of global
geometric analysis on singular K\"ahler spaces including the spectral theorem,
heat kernel estimates, eigenvalue estimates and diameter estimates. Such
estimates were only known previously in very special cases such as Bergman
metrics. As a consequence, we derive various geometric estimates, such as the
diameter estimate and the Sobolev inequality, for K\"ahler-Einstein currents on
projective varieties with definite or vanishing first Chern class.
