Ir al contenido principal

## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. Write the details of the proof of Vitalli's Lemma

2. Use the Hardy-Littlewood maximal theorem to prove the Lebesgue differentiation theorem: if $f\in L^1(\R^d)$, then $\displaystyle \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y) dy = f(x)$ for almost every $x\in\R^d.$

3. State conditions on $\Phi$ such that, for $f\in L^1(\R^d)$, $\displaystyle \lim_{t\to 0} \Phi_t*f(x) = f(x)$ for almost every $x\in\R^d.$

4. Let $f(x) = \begin{cases} \dfrac{1}{|x|(\log|x|)^2} & 0<|x|<1/2\\0&\text{otherwise.}\end{cases}$
Then $f\in L^1(\R)$, but $Mf$ is not locally integrable.