Sufficient conditions for boundedness of convolution operators in rearrangement-invariant spaces

We prove equivalence of sufficient conditions for boundedness of convolution integral operators in rearrangement-invariant spaces which were obtained in the articles of S. G. Krein, E. M. Semenov, and the author. The first of these conditions generalizes the Hardy–Littlewood–Sobolev condition and the second is a modification of the Hörmander condition.