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

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