Barbuti derived some condition for the boundedness of solutions of equation (1) (see article) and showed the relation of this condition to the results of Caccioppoli and Gusarov. Barbuti's results are interesting, but the formulation of the main theorem and the proof are rather cumbersome. Meanwhile, the main proofs can be carried out in a very elementary way, using the usual deductions that are always used in the study of equation (1); they are available, for example, in B. M. Levitan's book “Decomposition by Eigenfunctions".In the present note we give simplified proofs of some results of Barbuti and Gusarov.