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In this paper, we give the following dominated theorem: Let φ(g) ∈L1(G//K), φε(t)= 1/ε sh t/ε/sht φ(t/ε), ε＞ 0, and the least radical decreasing dominated y≥t≥0on (0, ∞), then for any f ∈ L1 loc(G//K), the following inequality holds:sup |φε* f(x)|≤Cmf(x),ε＞0where mf(x) is the Hardy-Littlewood maximal function of f, and C=‖φ‖1.An application of this dominated theorem is also given.