数学代写|EE261 Fourier analysis

The dyadic cubes in $\mathbb{R}^d$ are the sets of the form
$$
Q_{n, k}=\left[k_1 2^n,\left(k_1+1\right) 2^n\right) \times \cdots \times\left[k_d 2^n,\left(k_d+1\right) 2^n\right)
$$
were $n$ ranges over $\mathbb{Z}$ and $k \in \mathbb{Z}^d$.
(a) Given a collection of dyadic cubes whose diameters are bounded, show that one may find a sub-collection which covers the same region of $\mathbb{R}^d$ but with all cubes disjoint.
(b) Define the (uncentered) dyadic maximal function by
$$
\leftM_D f\right=\sup _{Q \ni x} \frac{1}{|Q|} \int_Q f(y) d y
$$
where the supremum is over all dyadic cubes that contain $x$. Show that this operator is of weak type $(1,1)$.
(c) Deduce boundedness of the Hardy-Littlewood maximal function from the above.

Remarks: Part (a) provides a replacement for the Vitali Covering Lemma. I propose you address (c) ‘geometrically’; draw some pictures in the planar $(d=2)$ case.

(a) Evaluate
$$
D_N(x)=\sum_{n=-N}^N e^{2 \pi i n x}
$$
and show that it is not an approximate identity on $\mathbb{T}$.
(b) Show that $\frac{1}{2 N+1}\left|D_N(x)\right|^2$ is an approximate identity and derive its relation to the Fejer kernel.
(c) Calculate
$$
\sum_{n \in \mathbb{Z}} r^{|n|} e^{2 \pi i n x}
$$
for $0<r<1$ and show that for $r \rightarrow 1$ it gives rise to an approximate identity.
(d) Suppose $\phi_n$ is an approximate identity and $d \mu$, a finite complex measure on $\mathbb{T}$.
Show that $\phi_n * d \mu$ converges weak-* to $d \mu$.
Note: $d \mu_n$ converges weak-* to $d \mu$ iff for every bounded continuous function, $f$, $\int f d \mu_n \rightarrow \int f d \mu$

(a) Given $f \in L^p(\mathbb{R}), 1 \leq p<\infty$, show that $t \mapsto f(x+t)$ defines a continuous map of $\mathbb{R}$ into $L^p(\mathbb{R}, d x)$.
(b) Show that it is not equi-continuous as $f$ varies over the set of $f$ with $|f|_{L^p} \leq 1$.
(That is, $\epsilon$ cannot be chosen from $\delta$ independently of $f$.)
(c) Show that part (a) is false for $L^{\infty}$ and $M(\mathbb{R})$.

(From Wolff §4.) Find a sequence of Schwartz functions $\phi_n$ such that (a) $\left|\phi_n\right|_{L^p}$ and $\left|\hat{\phi}n\right|{L^{p^{\prime}}}$ are constant. The supports of $\hat{\phi}n$ are disjoint and those of $\phi_n$ are almost disjoint. Use $\sum{n=1}^N \phi_n$ to show that if $|\hat{f}|_{L^{p^{\prime}}} \lesssim|f|_{L^p}$ then $p \leq 2$.

By almost disjoint we mean $\left|\sum_{n=1}^N \phi_n\right|_{L^p}^p \leq \frac{100}{99} \sum_{n=1}^N\left|\phi_n\right|_{L^p}^p$. Notice that if the supports were actually disjoint, then $100 / 99$ could be replaced by 1 .
Hint: Take a single $C_c^{\infty}$ function and modify it by translation and multiplication by characters.