Carleman estimates for Baouendi–Grushin operators with applications to quantitative uniqueness and strong unique continuation

ABSTRACT

In this paper, we establish some new $L^2-L^2$ Carleman estimates for the Baouendi–Grushin operators ${\mathcal{B}}_{\gamma }$, in Equation (1). We apply such estimates to obtain: (i) an extension of the Bourgain–Kenig quantitative unique continuation and (ii) the strong unique continuation property for some degenerate sublinear equations.

Keywords:

• Baouendi–Grushin operators
• strong unique continuation

2010 Mathematics Subject Classifications:

• 35H20
• 35B60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

First author is supported by SERB Matrix grant MTR/2018/000267. Second author is supported in part by a Progetto SID (Investimento Strategico di Dipartimento) ‘Non-local operators in geometry and in free boundary problems, and their connection with the applied sciences’, University of Padova, 2017. Third author is supported by SERB National Postdoctoral fellowship, PDF/2017/002780.