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In this paper we investigate Besov spaces on graded Lie groups. We prove aNikolskii type inequality (or the Reverse H\"older inequality) on graded Liegroups and as consequence we obtain embeddings of Besov spaces. We prove aversion of the Littlewood-Paley theorem on graded Lie groups. The results areapplied to obtain embedding properties of Besov spaces and multiplier theoremsfor both spectral and Fourier multipliers in Besov spaces on graded Lie groups.In particular, we give a number of sufficient conditions for the boundedness ofFourier multipliers in Besov spaces.

Littlewood-Paley theorem, Nikolskii inequality, Besov spaces, Fourier and spectral multipliers on graded Lie groups