Problem Set 5

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

    1. A polynomial p\in\mathscr P_k if and only if \sum x_j \partial_j p = kp.
    2. \dim \mathscr P_k = \binom{k+d-1}{k}
  1. Let p\in\mathscr P_k, and consider its orthogonal projection |x|^2q onto the space |x|^2\mathscr P_{k-2} with respect to the inner product seen in class. Then p - |x|^2q is harmonic. (Hint: Prove \langle r,\Delta(p - |x|^2q\rangle = 0 for every r\in\mathscr P_{k-2}.)
  2. If p\in\mathscr H_k, then \dfrac{p}{|x|^{2k+d-2}} is harmonic on \R^d\setminus\{0\}.
  3. The spaces are invariant under rotations , i.e. if f\in H_k then f\circ R\in H_k for any rotation R.
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