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Problem set 11, Harmonic Analysis

Due May 3

Problem 1

Let $latex \gamma$ be the lower semicircle of radius $latex N$ around the origin, and $latex \xi > 0.$ Then
$latex \displaystyle \int_\gamma f(z) dz = 2\pi i\text{Res}_{z=-it} f(z) = i e^{-2\pi t\xi},$
where $latex f(z)$ is the function defined by
$latex f(z) = \dfrac{1}{\pi} \dfrac{z}{z^2+t^2} e^{-2\pi iz\xi}.$

Problem 2

If $latex f\in L^1(\mathbb R)$ and diferentiable at $latex x\in\mathbb R$, then the limit
$latex \displaystyle \lim_{t\to 0} \int_{|y|\ge t} \frac{f(x-y)}{y} dy$
exists. (Hint: Use the identity, for any $latex \delta_n>0$,
$latex \displaystyle \int_{t\le|y|<\delta_n} \frac{f(x-y)}{y} dy = \int_{t\le|y|<\delta_n} \frac{f(x-y) - f(x)}{y} dy + \int_{t\le|y|<\delta_n} \frac{f(x)}{y} dy,$
and take $latex \delta_n\to 0$.)

Problem 3

If $latex f_n, g_n$ are sequences in $latex C_c^\infty(\mathbb R)$ that converge in $latex L^1(\mathbb R)$ to $latex f$, then $latex Hf_n$ and $latex Hg_n$ converge in measure to the same function $latex Hf$.

Problem 4

 Let $latex K\in L^2(\mathbb R)$ satisfy
  1. $latex \hat K \in L^\infty(\mathbb R)$; and
  2. there exists a constant $latex A>0$ such that $latex |K'(x)| \le \dfrac{A}{|x|^2}$ for any $latex x\in\mathbb R$.
Then the operator $latex Tf = K*f$ is weakly bounded on $latex L^1(\mathbb R)$.

Problem 5

We can replace hypothesis (2) from the previous problem by
  • there exists a constant $latex A>0$ such that $latex \displaystyle \int_{|x|\ge 2|y|} | K(x - y) - K(x)| dx \le A$ for any $latex y\in\mathbb R$.