  In this paper, we obtain new Carleman estimates for a class of variable
coefficient degenerate elliptic operators whose constant coefficient model at
one point is the so called Baouendi-Grushin operator. This generalizes the
results obtained by the two of us with Garofalo in [9] where similar estimates
were established for the "constant coefficient" Baouendi-Grushin operator.
Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness
result in the variable coefficient setting; (ii) and a strong unique
continuation property for a class of degenerate sublinear equations. We also
derive a subelliptic version of a scaling critical Carleman estimate proven by
Regbaoui in the Euclidean setting using which we deduce a new unique
continuation result in the case of scaling critical Hardy type potentials.
