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## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. The function $x\mapsto P(x,\xi)$ is harmonic for each $\xi$, where $P(x,\xi)$ is the Poisson kernel for the ball.

2. (Symmetry Lemma) For any $x\in\mathbb B$ and $\xi\in\mathbb S$, $\displaystyle \Big| |x|\xi - \frac{x}{|x|}\Big| = |x-\xi|$.

3. (Harnack's inequality) If $u$ is a positive harmonic function on $\bar{\mathbb B}$, then
$\displaystyle \frac{1-|x|}{(1+|x|)^{d-1}} u(0) \le u(x) \le \frac{1 + |x|}{(1-|x|)^{d-1}}u(0).$

1. (Hopf Lemma) If $u$ is a nonconstant harmonic function on $\bar{\mathbb B}$ and attains its maximum at $\zeta\in\mathbb S$, there exists $c>0$ such that $u(\zeta) - u(r\zeta) \ge c(1-r)$ for any $0 < r < 1.$

2. If $u$ is harmonic on $\bar{\mathbb B}$ and its normal derivative is 0 everywhere on $\mathbb S$, then $u$ is constant.