Problem Set 4

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

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


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


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


4. 
   
   
   
   1. (Hopf Lemma) If $latex u$ is a nonconstant harmonic function on $latex \bar{\mathbb B}$ and attains its maximum at $latex \zeta\in\mathbb S$, there exists $latex c>0$ such that $latex u(\zeta) - u(r\zeta) \ge c(1-r)$ for any $latex 0 < r < 1.$
   
   
   2. If $latex u$ is harmonic on $latex \bar{\mathbb B}$ and its normal derivative is 0 everywhere on $latex \mathbb S$, then $latex u$ is constant.