Improved quantitative unique continuation for complex-valued drift equations in the plane

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form Δu + W · ∇u = 0 in ℝ2, where W = W1 + iW2 with each Wj being real-valued. Under the assumptions that Wj ∈ Lqj for some q1 ∈ [2, ∞], q2 ∈ [2, ∞] and that W2 exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.