Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

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 [ 10 ] 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.