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## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. Let $P:\R^d\to\R^d$ be orthogonal.

1. $\Delta(u\circ P) = (\Delta u)\circ P$

2. If $u$ is harmonic, then $u\circ P$ is also harmonic.

2. If $T$ is a translation and $u$ is harmonic, then $u\circ T$ is harmonic.

1. Use polar coordinates to verify $\displaystyle \int_{\R^2}e^{-\pi |x|^2} dx = 1.$

2. Use (3.1) to prove $\displaystyle \int_{\R^d} e^{-\pi |x|^2} dx = 1$, for any $d\ge 1$.

3. Use spherical coordinates and the previous result to prove $\omega_d = \dfrac{2\pi^{d/2}}{\Gamma(d/2)}$, where $\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t} dt$ is the Gamma function.

3. Use the divergence theorem $\displaystyle \int_\Omega \nabla\cdot \vec F dx = \int_{\partial\Omega} \vec F \cdot \hat{\mathbf n} \; d\sigma$ to prove Green's identity. (Hint: Consider $\vec F = u \nabla v - v\nabla u$.)

4. The function $v(x) = \begin{cases} \log |x| & d=2\\ |x|^{2-d} & d>2\end{cases}$ is harmonic in $\R^d\setminus\{0\}$.