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 $$\Delta^2_{X} u = V u,$$ where $\Delta_{X} = \Delta_{x} + |x|^{2\beta} \Delta_{y}$ ($0 < \beta \leq 1$), 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 $|Z V| \leq K \psi$ for some constant $K>0$, where $Z= \sum_{i=1}^m x_i \partial_{x_i} + (\beta+1)\sum_{j=1}^n y_j \partial_{y_j}$, $\psi$ is the angle function given by $\psi = \frac{|x|^{2\beta}}{\rho^{2\beta}}$, and $$\rho(x,y) = \left(|x|^{2(\beta+1)} + (\beta+1)^2 |y|^2\right)^{\frac{1}{2(\beta+1)}}$$ 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.