User:Tkojar/Sandbox/Hardy-Littlewood Maximal Inequality

Theorem
Theorem (Weak Type Estimate). For $d\geq 1$ and $f\in L^{1}(\mathbb{R}^{d})$ there exists $C_{d}>0,\lambda>0$ such that the Hardy-Littlewood maximal function Mf satisfies


 * $\displaystyle \left |\{Mf > \lambda\} \right |< \frac{C_d}{\lambda} \Vert f\Vert_{L^1 (\mathbf{R}^d)}.$

Theorem (Strong Type Estimate) For $d\geq 1$, $p\in (1,\infty]$ and $f\in L^{p}(\mathbb{R}^{d})$ there is a constant $C_{p,d}>0$ such that the Hardy-Littlewood maximal function Mf satisfies


 * $\displaystyle \Vert Mf\Vert_{L^p (\mathbf{R}^d)}\leq C_{p,d}\Vert f\Vert_{L^p(\mathbf{R}^d)}$

Proof
For $p=\infty$, the inequality is trivial (since the average of a function is no larger than its essential supremum).

For $1\leq p<\infty$, first we shall use the following version of the Vitali Covering Lemma to prove the weak-type estimate.

Vitali Covering Lemma: Let X be a separable metric space and $\mathcal{F}$ family of open balls with bounded diameter.

Then $\mathcal{F}$ has a countable subfamily $\mathcal{F}'$ consisting of disjoint balls such that


 * $\displaystyle \bigcup_{B \in \mathcal{F}} B \subset \bigcup_{B \in \mathcal{F'}} 5B$

where 5B is B with 5 times radius.

If $Mf(x)>t$ then, by definition, we can find a ball $B_{x}$ centered at x such that


 * $\displaystyle \int_{B_x} |f|dy > t|B_x|.$

By the lemma, we can find, among such balls, a sequence of disjoint balls $B_{j}$ such that the union of $5B_{j}$ covers $\{x: Mf(x)>t\}$.

It follows:


 * $|\{Mf > t\}| \le 5^d \sum_j |B_j| \le {5^d \over t} \int |f|dy.$

This completes the proof of the weak-type estimate.

We next deduce from this the $L^{p}$ bounds. Define function b(x) by $b(x) = f(x)$ if $|f(x)| > t/2$ and $b(x)=0$ otherwise.

By the weak-type estimate applied to the function b, we have:


 * $\displaystyle |\{Mf > t\}| \le {2C \over t} \int_{|f| > \frac{t}{2}} |f|dx, $

with $C:=5^{d}$. Then


 * $\displaystyle \|Mf\|_p^p = \int \int_0^{Mf(x)} pt^{p-1} dt dx = p \int_0^\infty t^{p-1} |\{ Mf > t \}| dt$

By the estimate above we have:


 * $\displaystyle \|Mf\|_p^p \leq p \int_0^\infty t^{p-1} \left ({2C \over t} \int_{|f| > \frac{t}{2}} |f|dx \right ) dt = 2C p \int_0^\infty \int_{|f| > \frac{t}{2}} t^{p-2} |f| dx dt = C_p \|f\|_p^p$

where the constant $C_{p}$ depends only on p and d. This completes the proof of the theorem.