Abstract Let ( 𝒳 , d , μ ) $({\mathcal {X}},d,\mu )$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. Under this assumption, the authors establish some weighted norm inequalities for the multilinear Calderón–Zygmund operators via first obtaining some weighted estimates involving the John–Strömberg maximal operators and the John–Strömberg sharp maximal operators.