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Let μ be a nonnegative Radon measure on â„Âd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r))≤Crn for all x∈â„Âd, r>0 and some fixed n∈(0,d]. In this paper, the authors prove that for suitable indexes àand λ, the parametrized gλ∗ function ℳλ∗,àis bounded on Lp(μ) for p∈[2,∞) with the assumption that the kernel of the operator ℳλ∗,àsatisfies some Hörmander-type condition, and is bounded from L1(μ) into weak L1(μ) with the assumption that the kernel satisfies certain slightly stronger Hörmander-type condition. As a corollary, ℳλ∗,àwith the kernel satisfying the above stronger Hörmander-type condition is bounded on Lp(μ) for p∈(1,2). Moreover, the authors prove that for suitable indexes àand λ,ℳλ∗,àis bounded from L∞(μ) into RBLO(μ) (the space of regular bounded lower oscillation functions) if the kernel satisfies the Hörmander-type condition, and from the Hardy space H1(μ) into L1(μ) if the kernel satisfies the above stronger Hörmander-type condition. The corresponding properties for the parametrized area integral ℳSàare also established in this paper.