A unique continuation property for u vert ≤ V u vert
Let uΩ C^n → C^m, for n ≥ 2 and m ≥ 1. Let 1 ≤ p ≤ 2, and 2(2n)^2 -1 ≤ q < ∞ such that 1p + 1p’ = 1 and 1p - 1p’ = 1q. Suppose u vert ≤ V u vert , where V L^q_loc (Ω ). Then u has a unique continuation property in the following sense: if u W^1,p_loc (Ω ) and for some z_0 Ω , u Vert _L^p’(B(z_0,r)) decays faster than any powers of r as r → 0, then u 0. The same result holds for q=∞ if u is scalar-valued (m=1) , number=1, journal=MATHEMATICA SCANDINAVICA, author=Shi, Ziming, year=2025, month=Mar.