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

We investigate the quantitative unique continuation property for solutions to $${\mathrm{\Delta }}_X^{2}u=Vu,$$ where ${\mathrm{\Delta }}_X={\mathrm{\Delta }}_x+\mathrm{\mid }x{\mathrm{\mid }}^{2\beta }{\mathrm{\Delta }}_y$ ($0<\beta \le 1$), 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 $\mathrm{\mid }ZV\mathrm{\mid }\le K\psi$ for some constant $K>0$, where $Z={\sum }_{i=1}^m{x}_i{\mathrm{\partial }}_{x_i}+(\beta +1){\sum }_{j=1}^n{y}_j{\mathrm{\partial }}_{y_j}$, $\psi$ is the angle function given by $\psi =\frac{\mathrm{\mid }x{\mathrm{\mid }}^{2\beta }}{{\rho }^{2\beta }}$, and $$\rho (x,y)={(\mathrm{\mid }x{\mathrm{\mid }}^{2(\beta +1)}+(\beta +1{)}^2\mathrm{\mid }y{\mathrm{\mid }}^2)}^{\frac{1}{2(\beta +1)}}$$ 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.