The invention relates to a process for preparing an alkanone and/or alkanol by oxidizing an alkane with 3-30 C-atoms, using oxygen, to form an alkyl hydroperoxide, followed by a decomposition of the resulting alkyl hydroperoxide in the presence of a metal compound immobilized on a carrier, which carrier carries aliphatic or aromatic amine groups or sulphide groups. The process is preferably applied to cycloalkanes.