Unique Continuation for Degenerate Elliptic Equations

A famous result, first proved in ℝ2 by Carleman [C] in 1939, states that if $$V \in L_{\text{loc}}^{\infty}(\mathbb{R}^N)$$ and u is a solution to Δu = Vu in a connected open set $$D \subset \mathbb{R}^N$$, then u cannot vanish to infinite order at a point x₀ ∈ D unless u ≡ 0 in D. We are interested in analogous results when the (elliptic) Laplacian in ℝN is replaced by a subelliptic operator of the type (1.1)$$\mathcal{L} = \sum_{j=1}^{N-1}X_j^2$$, where X₁,…, X_N−1 are smooth vector fields satisfying Hörmander’s condition for hypoellipticity [H]: $$\text{rank Lie} [X_1,\cdots, X_{N-1}] = N \text{ at every point}$$.