Hardy–Littlewood Maximal Inequality

• categories: Measure Theory, Real Analysis, Theorem

The Hardy–Littlewood maximal inequality provides a bound on the measure of the set where the Hardy–Littlewood maximal function exceeds a given threshold, in terms of the $L^1$ norm of the original function.

Statement :

Let $f\in L^1({\mathbb{R}}^n)$ (locally integrable function on ${\mathbb{R}}^n$). Then there exists a constant $C_n>0$, depending only on the dimension $n$, such that for any $\lambda >0$:

$$|\{x\in {\mathbb{R}}^n:Mf(x)>\lambda \}|\le \frac{C_n}{\lambda }{\int }_{{\mathbb{R}}^n}|f(y)|dy,$$

where $Mf$ is the Hardy–Littlewood Maximal Function

Proof Outline:

The proof relies on the Vitali covering lemma and some properties of the maximal function.

1. Decomposition of the set: Consider the set $A_{\lambda }=\{x\in {\mathbb{R}}^n:Mf(x)>\lambda \}$. By the definition of $Mf(x)$, for each $x\in A_{\lambda }$, there is a ball $B_x$ centered at $x$ such that:

   $$\frac{1}{|B_x|}{\int }_{B_x}|f(y)|dy>\lambda .$$

2. Vitali covering lemma: Use the Vitali covering lemma to select a countable, disjoint subcollection of these balls $\{B_i\}$ that still cover most of the set $A_{\lambda }$ up to a set of measure zero.

3. Estimate using the disjoint balls: For each ball $B_i$, we know:

   $$\lambda |B_i|<{\int }_{B_i}|f(y)|dy.$$

   Summing over the disjoint collection $\{B_i\}$, we get:

   $$\lambda \sum _i|B_i|<\sum _i{\int }_{B_i}|f(y)|dy\le {\int }_{{\mathbb{R}}^n}|f(y)|dy.$$

4. Conclusion: Since $\{B_i\}$ are disjoint and cover most of $A_{\lambda }$, we have:

   $$|A_{\lambda }|\le \sum _i|B_i|\le \frac{1}{\lambda }{\int }_{{\mathbb{R}}^n}|f(y)|dy.$$

Thus, the measure of the set where $Mf(x)>\lambda$ is controlled by $\frac{1}{\lambda }$ times the $L^1$ norm of $f$.