Carleman estimates for higher step Grushin operators

The higher step Grushin operators ${\Delta }_{\alpha }$ are a family of sub-elliptic operators which degenerate on a sub-manifold of $R^{n+m}$. This paper establishes Carleman-type inequalities for these operators. It is achieved by deriving a weighted $L^p-L^q$ estimate for the Grushin-harmonic projector. The crucial ingredient in the proof is the addition formula for Gegenbauer polynomials due to T. Koornwinder and Y. Xu. As a consequence, we obtain the strong unique continuation property for the Schrödinger operators $-{\Delta }_{\alpha }+V$ at points of the degeneracy manifold, where V belongs to certain $L_{\mathrm{loc}}^r\left(R^{n+m}\right)$.