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Problem Set 2

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

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

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

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

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

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

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

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

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

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