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This paper constructs a class of martingale transforms based on L\'evyprocesses on Lie groups. From these, a natural class of bounded linearoperators on the $L^p$-spaces of the group (with respect to Haar measure) for$1<p<\infty$, are derived. On compact groups these operators yield Fouriermultipliers (in the Peter-Weyl sense) which include the second order Riesztransforms, imaginary powers of the Laplacian, and new classes of multipliersobtained by taking the L\'evy process to have conjugate invariant laws.Multipliers associated to subordination of the Brownian motion on the group arespecial cases of this last class. These results extend (and the proofssimplify) those obtained in \cite{BanBieBog, BanBog} for the case of $\bR^n$.An important feature of this work is the optimal nature of the $L^p$ bounds.

Martingale transform and L\'evy Processes on Lie Groups