Unique continuation with L^{2n} potentials in higher dimensions for vector-valued solutions

Determine whether strong unique continuation holds for vector-valued Sobolev solutions u: Q ⊂ C^n → C^N with n ≥ 3 and N ≥ 2, where u ∈ W^{1,p}(Q) for some p > 2n and satisfies the Schrödinger-type inequality |∂u| ≤ V|u| almost everywhere on Q with potential V ∈ L^{2n}(Q). Specifically, show that if u vanishes to infinite order in the L^1 sense at some point z₀ ∈ Q, then u must vanish identically on Q.

Background

The paper proves strong unique continuation for Sobolev solutions to the inequality |∂u| ≤ V|u| under several regimes: (i) for u ∈ W^{1,1}_{loc} when V ∈ L^p_{loc} with p > 2n; and (ii) for scalar solutions (N = 1) or two complex dimensions (n = 2) when V ∈ L^{2n}, provided u has slightly higher regularity u ∈ W^{1,2n+ε}_{loc}. These results extend the authors’ previous work from smooth to Sobolev solutions.

However, for V ∈ L^{2n} in higher dimensions with N ≥ 2, their slicing method breaks down because the complex radial restrictions of such potentials need not be locally integrable (as demonstrated by Example 3). The authors therefore pose whether strong unique continuation still holds under these critical integrability conditions. They note that resolving this question can be reduced to a weighted one-dimensional problem (Question 2).