Problem Set 2

Park City Mathematics Institute
Undergraduate 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\}.
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