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\begin{document}
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\large



\title{Math 533, Spring 2020, HW7} 
\author{Due in class, April 6}
\maketitle

\begin{problem} 
Suppose $f \in L^p(\bR)$. If there exists $h \in L^p(\bR)$ such that
\[
 \lim_{y \to 0} \|y^{-1}(f^{-y}-f) - h\|_p = 0,
\]
we call $h$ the \emph{strong $L^p$ derivative} of $f$ and write $h = df/dx$.  If $f \in L^p(\bR^n)$, $L^p$ derivatives of $f$ are defined similarly. If $p$ and $q$ are conjugate exponents, $f \in L^p$, $g \in L^q$, and the $L^p$ derivative $\partial_j f$ exists, then prove $\partial_j (f*g)$ exists in the ordinary sense and equals $(\partial_j f)*g$.
\end{problem}

\begin{problem}
 Let $\phi\in L^1(\bR^n)$ satisfy $|\phi(x)| \leq C (1 + |x|)^{-n - \epsilon}$ for some $C, \epsilon > 0$, and $\int \phi(x)dx = a.$ For $t >0$, $\phi_t(x)= t^{-n} \phi\left(\frac{x}{t} \right)$. If $f \in L^p$ define the $\phi$-maximal function of $f$ to be $M_\phi f(x) = \sup_{t>0} |f*\phi_t(x)|$. The \emph{Hardy-Littlewood maximal function} $Hf$ is $M_\phi|f|$ where $\phi$ is the characteristic function of the unit ball, divided by the volume of the ball. Show that there is a constant $C$, independent of $f$, such that $M_\phi f \leq CH f$.
\end{problem}

\begin{problem}
 Young's inequality shows that $L^1$ is a Banach algebra with convolution as multiplication.
 \begin{enumerate}
  \item If $\sI$ is an ideal in the algebra $L^1$, prove that its closure is, also.
  \item If $f \in L^1$, the smallest closed ideal in $L^1$ containing $f$ is the smallest closed subspace of $L^1$ containing translates of $f$.
 \end{enumerate}

\end{problem}


\begin{problem}
Show that if $f \in L^1(\bR^n)$, $f$ is continuous at 0, and $\hat{f} \geq 0$, then $\hat{f} \in L^1$.
\end{problem}

%\begin{problem}
% Let $f(x) = e^{-2\pi x}x^{\alpha-1}$ for $x>0$ and $f(x) = 0$ for $x \leq 0$, where $\alpha > 0$.  Then $f \in L^1$ and $\hat{f}(\xi) = \Gamma(\alpha)[(2\pi)(1 + i\xi)]^{-\alpha}$. Deduce from this that if $a > \frac{1}{2}$ and $b > \frac{1}{2}$, then
% \[
%  \int_{-\infty}^\infty (1 + ix)^{-a}(1 + ix)^{-b} dx = \frac{2^{2-a-b}\pi \Gamma(a + b-1)}{\Gamma(a)\Gamma(b)}.
% \]

%\end{problem}

%\begin{problem}
% (Wirtinger's inequality) If $f \in C^1([a,b])$ and $f(a) = f(b) = 0$, show that
% \[
%  \int_a^b |f(x)|^2 dx \leq \left(\frac{b-a}{\pi} \right)^2 \int_a^b |f'(x)|^2 dx.
% \]

%\end{problem}

%\begin{problem}
% Since $\sF$ commutes with rotations, the Fourier transform of a radial function is radial; that is, if $F \in L^1$ and $F(x) = f(|x|)$, then $\hat{F}(\xi) = g(|\xi|)$, where $f$ and $g$ are related as follows.
% \begin{enumerate}
%  \item Let $J(\xi) = \int_S e^{ix \cdot \xi} d\sigma(x)$ where $\sigma$ is surface measure on the unit sphere $S$ in $\bR^n$.  Then $J$ is radial, say $J(\xi) = j(|\xi|)$ and \[g(\rho) = \int_0^\infty j(2\pi r\rho) f(r)r^{n-1}dr.\]
%  \item $J$ satisfies $\sum_1^n \partial_k^2 J + J = 0$.
%  \item $j$ satisifies $\rho j''(\rho) + (n-1) j'(\rho) + \rho j(\rho) = 0$.
%  \item If $n = 3$, $j(\rho) = 4\pi (\sin \rho)/\rho.$
% \end{enumerate}

%\end{problem}

\begin{problem}
 Let $f$ be a function on $\bT^1$ and $A_r f$ the $r$th Abel mean of the Fourier series of $f$. Check that
 \begin{enumerate}
  \item $A_r f = f*P_r$ where $P_r(x) = \sum_{-\infty}^\infty r^{|k|} e^{2\pi ikx}$ is the Poisson kernel for $\bT^1$.
  \item $P_r(x) = \frac{1-r^2}{1 + r^2 -2r \cos 2\pi x}$.
 \end{enumerate}

\end{problem}

\begin{problem}
 Given $f \in L^1(\bT^1)$, let $S_m f(x) = \sum_{-m}^m \hat{f}(k)e^{2\pi i kx}$ and 
 \[
  \sigma_m f(x) = \sum_{-m}^m \hat{f}(k)\left(1 - \frac{|k|}{m+1} \right) e^{2\pi i kx}.
 \]
Prove the following.
\begin{enumerate}
 \item $\sigma_m f = \frac{1}{m+1} \sum_0^m S_k f$.
 \item If $D_k$ is the $k$th Dirichlet kernel, we have $\sigma_m f = f*F_m$ where $F_m = \frac{1}{m+1} \sum_0^m D_k$. $F_m$ is the $m$th Fej\'{e}r kernel on $\bT^1$.
 \item $F_m(x) = \frac{\sin^2 (m+1)\pi x}{(m+1)\sin^2 \pi x}.$
\end{enumerate}

\end{problem}

\begin{problem}
 Prove the following.
 \begin{enumerate}
  \item If $D_m$ is the $m$th Dirichlet kernel, $\|D_m\|_1 \to \infty$ as $m \to \infty$.
  \item The Fourier transform is not surjective from $L^1 (\bT^1)$ to $C_0(\bZ)$.
 \end{enumerate}

\end{problem}


\end{document}

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