On the maximum principle for parabolic equations with unbounded coefficients

Translation of the paper “Interpolation of linear spaces and maximum estimates for solutions to parabolic equations” published in Russian in the collected volume “Partial differential equations”, Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1987, 50–72. Original proofs are essentially simplified. Some gaps are fixed and some comments are added.