Boundedness of Riesz potentials in nonhomogeneous spaces

For a class of linear operators including Riesz potentials on R(d) with a non-negative Radon measure mu, which only satisfies some growth condition, the authors prove that their boundedness in Lebesgue spaces is equivalent to their boundedness in the Hardy space or certain weak type endpoint estimates, respectively. As an application, the authors obtain several new end estimates.