Problem Set 4

Park City Mathematics Institute
Undergraduate 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|.SymmetryLemma.png
  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.
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