Weighted one-dimensional reduction for L^{2n} potentials and vector-valued solutions

Establish strong unique continuation for vector-valued Sobolev solutions u: Q ⊂ C → C^N with n ≥ 3 and N ≥ 2, where u ∈ W^{1,2}(Q) and satisfies the weighted Schrödinger-type inequality |∂u| ≤ |z|^{-1} V|u| almost everywhere on Q with V ∈ L^{2n}(Q). Specifically, prove that if u vanishes to infinite order in the L^2 sense at 0 ∈ Q, then u must vanish identically on Q.

Background

The authors show that for L^{2n} potentials, strong unique continuation holds when N = 1 or n = 2 if u has slightly higher regularity, but the general case remains unsettled. Their slicing strategy converts higher-dimensional problems to one-dimensional ones when integrability and flatness assumptions pass to slices.

They note that resolving the full higher-dimensional, vector-valued case with V ∈ L^{2n} reduces to proving a one-dimensional weighted unique continuation statement of the form |∂u| ≤ |z|^{-1} V|u| with V ∈ L^{2n}. Success on this weighted problem would imply the higher-dimensional result (Question 1).