Problem Set 6

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

  1. Verify the integral \displaystyle \int_{\R^d}\frac{dx}{(|x|^2+1)^\frac{d+1}{2}} = \frac{\pi^\frac{d+1}{2}}{\Gamma(\frac{d+1}{2})}.
  2. Use Minkowski inequality to prove that, if K\in L^1(\R^d) and f\in L^p(\R^d), then ||K*f||_{L^p}\le ||K||_{L^1}||f||_{L^p}.
  3. Prove that, if f\in C_c(\R^d) and y\in\R^d, then ||f(\cdot - ty) - f||_{L^p} \to 0 as t\to 0.
  4. Let \Phi\in L^1(\R^d) with \int \Phi = 1, and \Phi_t(x) = t^{-d}\Phi(x/t). Then
    1. \int \Phi_t = 1 for all t>0.
    2. There exists some M>0 such that \int |\Phi_t| \le M for all t > 0.
    3. For each \delta>0, \displaystyle \int_{|x|\ge\delta} |\Phi_t(x)|dx \to 0 as t\to 0.
    4. If f\in C_c(\R^d), then \Phi_t*f(x) \to f(x) uniformly.
  5. State conditions on \Phi (as in the previous exercise) so that there exists A>0 such that, for any t>0 and f\in L^1(\R^d), |\Phi_t*f(x)| \le A Mf(x).
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