---
title: Singular Integrals
description: Singular integrals are integral transforms with kernels that have singularities. Learners will understand the Hilbert and Riesz transforms, Calderón-Zygmund theory, and how these operators behave on various function spaces.
category: mathematics
subcategory: harmonic-analysis
difficulty: beginner, intermediate, advanced
url: /subject/singular-integrals
---

# Singular Integrals

Singular integrals are integral transforms with kernels that have singularities. Learners will understand the Hilbert and Riesz transforms, Calderón-Zygmund theory, and how these operators behave on various function spaces.

## Available Resources

2 Books • 2 Websites • 1 Papers

## Papers

### 1. Singular Integrals: The Roles of Calderón and Zygmund

**Author:** Elias M. Stein

Free, short, and by the highest authority on the subject; it gives a learner the map of why each piece of CZ theory exists before they open a 600-page text, which no textbook chapter does as well. Stein's expository account of how Calderón and Zygmund built the real-variable theory of singular integrals, tracing the Hilbert transform, the Calderón-Zygmund decomposition, weak-type and Lp bounds, and the later extensions the theory made possible.

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

**Link:** https://www.ams.org/notices/199809/stein.pdf

**Tags:** singular-integrals, calderon-zygmund-theory, harmonic-analysis, hilbert-transform, history-of-mathematics

## Websites

### 1. Notes for Harmonic and Real Analysis (V4B5, Universität Bonn)

**Author:** Herbert Koch

Complete lecture notes from a Bonn graduate course. Chapter 5 treats singular integrals of Calderón-Zygmund type, stating and proving the CZ theorem with worked examples; earlier chapters build the Hilbert transform, later ones cover Hardy spaces, BMO and Littlewood-Paley theory.

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

**Link:** https://www.math.uni-bonn.de/ag/ana/WiSe1415/V4B5_notes.pdf

**Tags:** harmonic-analysis, singular-integrals, calderon-zygmund-theory, bmo-hardy-spaces, littlewood-paley-theory

### 2. 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. Fourier Analysis (Graduate Studies in Mathematics 29)

**Author:** Javier Duoandikoetxea (translated by David Cruz-Uribe)

Presents the real-variable methods of Calderón and Zygmund: the Hilbert and Riesz transforms, singular integral operators, multipliers, weighted inequalities, H1 and BMO, and the T1 theorem. Compact, roughly 220 pages, with exercises and notes.

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

**Link:** https://www.amazon.com/dp/0821821725?tag=edmonddante07-20

**Tags:** fourier-analysis, singular-integrals, calderon-zygmund-theory, weighted-inequalities, harmonic-analysis

### 2. Singular Integrals and Differentiability Properties of Functions

**Author:** Elias M. Stein

Stein's 1970 monograph that established the real-variable theory of singular integrals: Calderón-Zygmund decomposition, Lp boundedness, Riesz transforms, Littlewood-Paley theory, and Sobolev and Lipschitz spaces. Readers learn the core estimates and how they control differentiability of functions.

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

**Link:** https://www.amazon.com/dp/0691080798?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, harmonic-analysis

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