---
title: Maximal Functions
description: Maximal functions measure the local behavior of a function by taking averages over shrinking neighborhoods. Learners will understand the Hardy-Littlewood maximal operator and its role in proving pointwise convergence and bounding singular integrals.
category: mathematics
subcategory: harmonic-analysis
difficulty: beginner, intermediate, advanced
url: /subject/maximal-functions
---

# Maximal Functions

Maximal functions measure the local behavior of a function by taking averages over shrinking neighborhoods. Learners will understand the Hardy-Littlewood maximal operator and its role in proving pointwise convergence and bounding singular integrals.

## Available Resources

1 Books • 4 Websites

## Websites

### 1. 245A, Notes 5: Differentiation Theorems

**Author:** Terence Tao

Graduate real-analysis lecture notes published as a blog post, proving the Hardy-Littlewood maximal inequality with covering and rising-sun arguments, then deducing the Lebesgue differentiation theorem, absolute continuity and bounded variation results, with many embedded exercises.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/

**Tags:** hardy-littlewood-maximal-function, lebesgue-differentiation-theorem, real-analysis, bounded-variation, absolute-continuity

### 2. Maximal Functions in Analysis

**Author:** Robert Fefferman (scribe: Philip Ascher)

Eleven-page self-contained REU lecture notes building from the fundamental theorem of calculus to the Hardy-Littlewood maximal theorem, proving the Vitali covering lemma, the weak-type L1 bound, the Lp case, and an elementary introduction to Calderón-Zygmund theory.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.math.uchicago.edu/~may/VIGRE/VIGRE2005/Fefferman.pdf

**Tags:** hardy-littlewood-maximal-function, vitali-covering-lemma, calderon-zygmund-theory, harmonic-analysis, real-analysis

### 3. 247A Lecture Notes 3: The Hardy-Littlewood Maximal Inequality

**Author:** Terence Tao

Fifteen pages of Terence Tao's UCLA graduate harmonic analysis notes covering the Hardy-Littlewood maximal inequality: Vitali-type covering arguments, the weak-(1,1) bound, Marcinkiewicz interpolation to Lp, dyadic maximal functions, and the Lebesgue differentiation theorem.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://www.math.ucla.edu/~tao/247a.1.06f/notes3.pdf

**Tags:** hardy-littlewood-maximal-function, harmonic-analysis, marcinkiewicz-interpolation, dyadic-maximal-function, lebesgue-differentiation-theorem

### 4. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Books

### 1. Real Analysis: Measure Theory, Integration, and Hilbert Spaces

**Author:** Elias M. Stein, Rami Shakarchi

Volume III of the Princeton Lectures in Analysis; chapter 3 develops differentiation of the integral through the Vitali covering lemma, the Hardy-Littlewood maximal function, its weak-type bound, and the Lebesgue differentiation theorem, with exercises and problems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0691113866?tag=edmonddante07-20

**Tags:** measure-theory, lebesgue-integration, hilbert-spaces, hardy-littlewood-maximal-function, real-analysis

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