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.
$\Box$
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
- 2005: Elias M. Stein and Rami Shakarchi: Real Analysis: Measure Theory, Integration, and Hilbert Spaces, ISBN 0-691-11386-6