Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition, the authors prove that for a class of Zygmund operators and Lipschitz functions as examples, their boundedness in Lebesgue spaces or the Hardy space H1 (μ) is equivalent to some endpoint estimates satisfied by them.This result is new even when the underlying measure μ is the d-dimensional Lebesgue measure.