Let μ be a Radon measure on Rd which satisfies the growth condition only namely, there is a constant C > 0 such that for all x ∈ Rd,r > 0 and for some fixed 0 < n ≤d, μ(B(x, r)) ≤Crn, where B(x, r) is the ball centered at x and having radius r. In this paper, we first give a new atomic characterization of the Hardy space H1(μ) introduced by X. Tolsa. As applications of this new characterization, we establish the (H1(μ), L1,∞(μ)) estimate of the commutators generated by RBMO(μ) functions with the Calderon–Zygmund operators whose kernels satisfy only the size condition and a certain minimum regularity condition. Using this endpoint estimate and a new interpolation theorem for operators which is also established in this paper and has independent interest, we further obtain the Lp(μ) (1 < p < ∞) boundedness of these commutators.