Carleman estimates for higher step Grushin operators

2024/02/12 by De Bie, Hendrik, Lian, Pan · 1 citation
#35B60 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 35H20 #Secondary 33C45

paper · doi:10.48550/arxiv.2402.07348

Abstract

The higher step Grushin operators Δ_α 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 -Δ_α+V at points of the degeneracy manifold, where V belongs to certain L^r_{\rm loc}(ℝ^n+m).

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