Problem Set 7

Park City Mathematics Institute
Undergraduate 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 $latex f\in L^1(\R^d)$, then $latex \displaystyle \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y) dy = f(x)$ for almost every $latex x\in\R^d.$


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


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