Unless I'm misreading, there exist such an analogue. See equation 2.5 and theorem 3.1 of [Fractional Laplacian in Conformal Geometry][1] by Sun-Yung Alice Chang and María del Mar Gonzáles.


  [1]: http://www.pagines.ma1.upc.edu/~mgonzalez/Papers/Chang_Gonzalez.pdf