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## Park City Mathematics InstituteUndergraduate Summer School 2018

### Introduction to Harmonic Analysis

1. Extend the result that the Hilbert transform is of weak-type $L^1$ to any operator of the form $Tf = K*f$ where $K$ satisfies:

1. $\hat K\in L^\infty(\R)$ (say, taking the Fourier transform in the $L^2$ sense);

2. there exists a constant $A>0$ such that $|K'(x)| \le A/x^2$ for any $x\in\R, \; x\not=0$.

2. Show that we can replace (2) above by the condition
2'. There exists a constant $A>0$ such that $\displaystyle \int_{|x|\ge 2|t|} |K(x-t) - K(x)| dx \le A$ for any $t\in\R$.

3. (Chebyshev Inequality) If $f\in L^p$, for some $1 < p <\infty$, then $|\{x : |f(x)|>\alpha\}| < \dfrac{1}{\alpha^p} ||f||_{L^p}^p$

4. Prove that we can substitute the condition $\hat K\in L^\infty$ with the boundedness of $T$ on any $L^q(\R), \; q>1$, that is,
1'. there exists a constant $M>0$ such that $||Tf||_{L^q} \le M||f||_{L^q}$.