An Introduction to Singular Integrals

Singular integrals are (generally speaking) integral transforms of the form \(\int K(x,y) f(y) dy\) where the kernel function \(K(x,y)\) has a singularity whenever \(x = y\). A familiar example is the Hilbert transform, defined by \(K(x,y) = (1/\pi)/(x - y)\). Singular integrals are important in harmonic analysis, partial differential equations, and several other fields. They are divided generally into convolution types (where \(K(x,y) = k(x - y)\) for some function \(k\)) and non-convolution types.

This book covers several specific kinds of singular integrals, as well as related topics such as the Hardy–Littlewood maximal operator, BMO (bounded mean oscillation) spaces, a little bit of Hardy spaces, and interpolation of operators on \(L^p\) spaces. The book admits (or brags) in its first sentence that “The topics developed here have been standard knowledge for at least forty years.”

The book generally takes a concrete approach, focussing on specific singular integrals. It deals mostly with transforms on \(\mathbb{R}^n\), but there’s also a chapter showing how to apply these ideas to transforms defined on other groups such as the torus and the integers. The exercises are extensive and bring in many additional topics not covered in the narrative.

This book is a good, concise introduction to the classical theory, that covers a wide variety of topics in the exercises. Another good source for the modern theory of singular integrals is Grafakos’s two volume set Classical Fourier Analysis (has a chapter on convolution-type singular integrals) and Modern Fourier Analysis (has a chapter on non-convolution-type singular integrals).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.