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The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\\in L^1(R^d): \\|f\\|_{H^1(L_U)}:= \\|sup_{t>0} |K_{t,U} f| \\|_{L^1(R^d)} < \\infty. Assume that U=V+W, where V\\geq 0 satisfies the global Kato condition sup_{x\\in R^d} \\int_{R^d} V(y)|x-y|^{2-d} < \\infty. We prove that, under certain assumptions on W\\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. 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