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In this paper, we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $\mathcal {L}_s u=u^{\frac {Q+2s}{Q-2s}}$
in a homogeneous Lie group, where $\mathcal {L}_s$
represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.
