Failure of strong unique continuation for harmonic functions on RCD spaces

Abstract Unique continuation of harmonic functions on RCD {\operatorname{RCD}} space is a long-standing open problem, with little known even in the setting of Alexandrov spaces. In this paper, we establish the weak unique continuation theorem for harmonic functions on RCD ⁡ ( K , 2 ) {\operatorname{RCD}(K,2)} spaces and give a counterexample for strong unique continuation in the setting of RCD ⁡ ( K , N ) {\operatorname{RCD}(K,N)} space for any N ≥ 4 {N\geq 4} and any K ∈ ℝ {K\in\mathbb{R}} .