Hardy–Littlewood Maximal Function

• categories: Measure Theory, Definition, Harmonic Analysis

The Hardy–Littlewood maximal function is a key tool in harmonic analysis and is used to study the local behavior of functions by averaging them over neighborhoods. It is fundamental in modern analysis due to its ability to control the size and behavior of functions, especially in contexts where pointwise control is difficult

Definition:

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a locally integrable function (i.e., $f\in L_{\text{loc}}^1({\mathbb{R}}^n)$). The Hardy–Littlewood maximal function of $f$, denoted $Mf$, is defined as:

$$Mf(x)=\underset{r>0}{\sup }\frac{1}{|B(x,r)|}{\int }_{B(x,r)}|f(y)|dy,$$

where $B(x,r)$ is the ball centered at $x$ with radius $r$, $|B(x,r)|$ is theLebesgue Measure of the ball, and the integral is the average of $|f|$ over $B(x,r)$.

Intuition:

The maximal function $Mf(x)$ gives the largest possible local average of $|f|$ over balls centered at $x$ of varying radii. It captures the “worst-case” behavior of $f$ near $x$ by considering the largest average in neighborhoods around $x$. This is useful in controlling pointwise behavior of $f$ via its averages.

Key Properties:

1. Hardy–Littlewood Maximal Inequality

2. $L^p$ boundedness (for $p>1$): For $1<p\le \infty$, there exists a constant $C_p$ such that:

   $$\Vert Mf{\Vert }_{L^p({\mathbb{R}}^n)}\le C_p\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}.$$

   This means that $Mf$ is bounded in $L^p$ spaces for $p>1$.

3. Pointwise Control: The maximal function $Mf(x)$ provides a form of pointwise control over $f$. Specifically, the Hardy–Littlewood maximal theorem guarantees that, for a function $f\in L^p({\mathbb{R}}^n)$ with $p>1$, $Mf(x)$ is finite almost everywhere.

Applications:

• Differentiation of Integrals: The Hardy–Littlewood maximal function is used in the proof of the Lebesgue Differentiation Theorem
• Singular Integral Theory: The boundedness of maximal functions is a key step in establishing the boundedness of singular integrals, such as the Hilbert Transform or Riesz transform.
• Control of Oscillations: The maximal function helps to control the oscillation of functions by providing a bound on the size of the set where the function is “large.”