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Unique continuation of harmonic functions on $RCD$ space is a long-standingopen problem, with little known even in the setting of Alexandrov spaces. Inthis paper, we establish the weak unique continuation theorem for harmonicfunctions on $RCD(K,2)$ spaces and give a counterexample for strong uniquecontinuation in the setting of $  RCD(K,N)$ space for any $N\geq 4$ and any $K\in \mathbb{R}$.

Failure of strong unique continuation for harmonic functions on RCD Spaces