Unique Continuation Problem on RCD Spaces. I

TL;DR

This paper proves a weak unique continuation theorem for caloric functions on compact RCD(K,2) spaces, constructs examples on RCD(K,4) spaces where non-trivial solutions vanish to infinite order, and establishes frequency estimates, advancing understanding of unique continuation in metric measure spaces.

Contribution

It establishes the weak unique continuation theorem for caloric functions on RCD spaces and provides counterexamples and frequency estimates, extending classical PDE results to metric measure spaces.

Findings

Weak unique continuation theorem for caloric functions on RCD(K,2) spaces.

Existence of RCD(K,4) spaces with non-trivial eigenfunctions vanishing to infinite order.

Frequency estimates for eigenfunctions and caloric functions on the metric horn.

Abstract

In this note we establish the weak unique continuation theorem for caloric functions on compact $RCD(K,2)$ spaces and show that there exists an $RCD(K,4)$ 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.