Meda Inequality for Rearrangements of the Convolution on the Heisenberg Group and Some Applications

The Meda inequality for rearrangements of the convolution operator on the Heisenberg group ℍn is proved. By using the Meda inequality, an O'Neil-type inequality for the convolution is obtained. As applications of these results, some sufficient and necessary conditions for the boundedness of the fractional maximal operator MΩ,α and fractional integral operator IΩ,α with rough kernels in the spaces Lp(ℍn) are found. Finally, we give some comments on the extension of our results to the case of homogeneous groups