Carleman estimates for higher step Grushin operators

The higher step Grushin operators Δα are a family of sub-elliptic operators which degenerate on a sub-manifold of Rⁿ⁺ᵐ. This paper establishes Carleman-type inequalities for these operators. It is achieved by deriving a weighted Lᵖ-Lq 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 Schr\"odinger operators -Δα+V at points of the degeneracy manifold, where V belongs to certain Lʳ_ loc(Rⁿ⁺ᵐ).