On certain degenerate and singular elliptic PDEs I: Nondivergence form operators with unbounded drifts and applications to subelliptic equations

Since our results are shown to apply to the convex functions |x|p with p≥2 and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables. In the light of these applications, it follows that the Monge-Ampère quasi-metric structure can be regarded as an alternative to the usual Carnot-Carathéodory metric in the study of certain subelliptic PDEs.