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## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. If $u$ is harmonic in the connected domain $\Omega$ and is not constant, then $u(\Omega)$ is open in $\R$.

2. Suppose $\Omega$ is bounded and that its boundary $\partial\Omega$ is connected. If $u$ is harmonic in $\Omega$, then $u(\Omega)\subset u(\partial\Omega)$.

3. A radial harmonic function on $\mathbb B$ is constant.

4. A positive harmonic function on $\R^d$ is constant.