Fourier multipliers on graded Lie groups

We study the $L^p$-boundedness of Fourier multipliers defined on graded nilpotent Lie groups via their group Fourier transform. We show that H\"ormander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper.