Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential

Abstract

We investigate the quantitative unique continuation property for solutions to $$\hat{I}{”}_X^{2}u=Vu,$$ where $\hat{I}{”}_X=\hat{I}{”}_x+|x{|}^{2{\hat{I}}^2}\hat{I}{”}_y$ ($0<{\hat{I}}^2\le 1$), with $x\in {\mathbb{R}}^m$ and $y\in {\mathbb{R}}^n$, denotes a class of subelliptic operators of Baouendi-Grushin type. The potential $V$ is assumed to be bounded and satisfy $|ZV|\le K\ddot{I}ˆ$ for some constant $K>0$, where $Z={\sum }_{i=1}^m{x}_i{\partial }_{x_i}+({\hat{I}}^2+1){\sum }_{j=1}^n{y}_j{\partial }_{y_j}$, $\ddot{I}ˆ$ is the angle function given by $\ddot{I}ˆ=\frac{|x{|}^{2{\hat{I}}^2}}{\ddot{I}{}^{2{\hat{I}}^2}}$, and $$\ddot{I}(x,y)={\left(|x{|}^{2({\hat{I}}^2+1)}+({\hat{I}}^2+1{)}^2|y{|}^2\right)}^{\frac{1}{2({\hat{I}}^2+1)}}$$ defines the associated pseudo-gauge. By adapting Almgren's approach, we establish an almost monotonicity formula for the frequency function. As a consequence, we derive a quantitative unique continuation result for solutions to the fourth-order subelliptic equation.