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

TL;DR

This paper establishes a quantitative unique continuation property for solutions to a class of fourth-order subelliptic operators of Baouendi-Grushin type with a potential, using an adapted Almgren's frequency function approach.

Contribution

It introduces an almost monotonicity formula for the frequency function of these operators, leading to new quantitative unique continuation results.

Findings

Established an almost monotonicity formula for the frequency function.

Proved quantitative unique continuation for solutions of the operator.

Extended the approach to a class of fourth-order subelliptic operators.

Abstract

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.