To oxidize an organic substrate so as to control the deactivation of an imide compound as a catalyst and to obtain a lower order oxidation product such as an alcohol, etc., in high selectivity and yield by bringing the organic substrate into contact with molecular oxygen in the presence of a specific imide compound in an organic solvent under specific conditions.

(A) An organic substrate which is preferably a nonaromatic hydrocarbon (e.g. cyclohexane or the like) is brought into contact with (B) molecular oxygen in the presence of (C) an imide compound of the formula (R1 and R2 are each H, a halogen or the like; X is O or OH; (n) is 1-3) (e.g. N-hydroxyphthalimide or the like) or the component C and (D) 0-0.28 mol.% based on the component A of a metal compound [e.g. cobalt acetate (II) tetrahydrate or the like] in (E) an organic solvent such as preferably a nitrile (e.g. acetonitrile or the like) at 10-85°C preferably under normal pressure or under pressure from about 2-70 atmosphere, for example, for 30 minutes to 48 hours to oxidize the component A.