User:Tkojar/Sandbox/Hardy-Littlewood Maximal Inequality

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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


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


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


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


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


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


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


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


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:


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


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


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


By the estimate above we have:


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


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

$\blacksquare$


Source of Name

This entry was named for Godfrey Harold Hardy and John Edensor Littlewood.


Sources