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## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. Let $\{\Phi_t\}_{t>0}$ be a collection that satisfies, for some constants $c_0, c_1, c_2$, and all $x\in\R^d$ and $t>0$,

• $\int \Phi_t = c_0$,

• $|\Phi_t(x)| \le c_1 t^{-d}$,

• $|\Phi_t(x)| \le c_2 t/|x|^{d+1}$.

Then there exists $A>0$ such that $|\Phi_t*f(x)|\le A Mf(x)$, for all $x\in\R^d$, and thus $\Phi_t*f(x)\to f(x)$ as $t\to 0$ for almost every $x\in\R^d.$

2. (Needs complex analysis) $\displaystyle \hat Q _y(\xi) = \lim_{N\to\infty}\frac{1}{\pi} \int_{-N}^N \frac{x e^{-2\pi ix\xi}}{x^2+y^2} dx = -i\text{sgn}\xi e^{-2\pi y |\xi|}$. (Hint: Use the residue theorem from complex analysis.)

3. (Needs complex analysis) Give another proof that the Hilbert transform is bounded on $L^2$ by considering the Cauchy integral $\displaystyle Cf(z) = \frac{1}{i\pi} \int_{-\infty}^\infty \frac{f(t)}{t-z} dt$.

1. If $f\in C_c(\R)$, $Cf(z) = O(1/|z|)$.

2. If $F(z) = Cf(z)$, then $\displaystyle \int_{-\infty}^\infty F(x)^2 dx = 0$

3. Calculate $\Re (F(x)^2)$, and conclude $||Hf||_{L^2} = ||f||_{L^2}.$