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Problem Set 5

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

    1. A polynomial $latex p\in\mathscr P_k$ if and only if $latex \sum x_j \partial_j p = kp.$

    2. $latex \dim \mathscr P_k = \binom{k+d-1}{k}$

  1. Let $latex p\in\mathscr P_k$, and consider its orthogonal projection $latex |x|^2q$ onto the space $latex |x|^2\mathscr P_{k-2}$ with respect to the inner product seen in class. Then $latex p - |x|^2q$ is harmonic. (Hint: Prove $latex \langle r,\Delta(p - |x|^2q\rangle = 0$ for every $latex r\in\mathscr P_{k-2}$.)

  2. If $latex p\in\mathscr H_k$, then $latex \dfrac{p}{|x|^{2k+d-2}}$ is harmonic on $latex \R^d\setminus\{0\}$.

  3. The spaces are invariant under rotations , i.e. if $latex f\in H_k$ then $latex f\circ R\in H_k$ for any rotation $latex R$.