Unique continuation problem on RCD Spaces. I

Abstract

In this note we establish the weak unique continuation theorem for caloric functions on compact <jats:italic>RCD</jats:italic>(<jats:italic>K</jats:italic>, 2) spaces and show that there exists an <jats:italic>RCD</jats:italic>(<jats:italic>K</jats:italic>, 4) space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point . We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.