Order-sharp estimates for decreasing rearrangements of convolutions

In this paper, we study estimates for convolutions on some classes of measurable, positive and radial symmetrical functions. On this base we prove then order-sharp estimates for decreasing and symmetrical rearrangements of convolutions and for weighted mean values of rearrangements. These estimates give, in particular, a reversal of the well-known inequalities for convolutions proved by R. O’Neil.