On the strong unique continuation property of a degenerate elliptic operator with Hardy-type potential

  • Agnid BanerjeeEmail author
  • Arka Mallick


In this paper, we prove the strong unique continuation property for the following degenerate elliptic equation
$$\begin{aligned} \Delta _zu +|z|^2\partial _t^2u = Vu,\quad (z,t) \in {\mathbb {R}}^N \times {\mathbb {R}} \end{aligned}$$
where the potential V satisfies either of the following growth assumptions
$$\begin{aligned}&\left| V(z,t) \right| \le \frac{f(\rho (z,t))}{\rho (z,t)^2},\ \text {where} \rho \text { is as in }(2.1)\text { and }\nonumber \\&\quad f\text { satisfies the Dini integrability condition as in }(1.3). \end{aligned}$$
or when
$$\begin{aligned}&\left| V(z,t) \right| \le C \frac{\psi (z,t)^{\epsilon }}{\rho (z,t)^2},\ \text {for some }\epsilon >0\text { with }\psi \text { as in }(2.6)\text { and } N\text { even.} \end{aligned}$$
This extends some of the previous results obtained in [18] for this subfamily of Baouendi–Grushin operators. As corollaries, we obtain new unique continuation properties for solutions u to
$$\begin{aligned} \Delta _{{\mathbb {H}}} u = Vu \end{aligned}$$
with certain symmetries as expressed in (1.6) where \(\Delta _{{\mathbb {H}}}\) corresponds to the sub-Laplacian on the Heisenberg group \({\mathbb {H}}^n\).


Unique continuation Baouendi–Grushin operator Hardy-type potential Carleman estimates 

Mathematics Subject Classification

35J70 35J75 



One of us, A.B., would like to thank Prof. Nicola Garofalo for several suggestions and feedbacks related to this work.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.TIFR Centre for Applicable MathematicsBangaloreIndia
  2. 2.Department of MathematicsEPFL SB CAMALausanneSwitzerland

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