# Problem Set 6

## Park City Mathematics InstituteUndergraduate 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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