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Singular integral

Functions in harmonic analysis mathematics

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Source updated April 30, 2026

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator

$T(f)(x)=\int K(x,y)f(y)\,dy,$

whose kernel function $K:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R}$ is singular along the diagonal $x=y$ . Specifically, the singularity is such that $|K(x,y)|$ is of size $|x-y|^{-n}$ asymptotically as $|x-y|\to 0$ . Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over $|y-x|\to \epsilon$ as $\epsilon \to 0$ , but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp spaces, for example $L^{p}(\mathbb {R} ^{n})$ .

01The Hilbert transform

The archetypal singular integral operator is the Hilbert transform $H$ . It is given by convolution against the kernel $K(x)=1/(\pi x)$ for $x$ in $\mathbb {R}$ . More precisely,

$H(f)(x)={\frac {1}{\pi }}\lim _{\varepsilon \to 0}\int _{|x-y|>\varepsilon }{\frac {1}{x-y}}f(y)\,dy.$

The most straightforward higher dimension analogues of these are the Riesz transforms, which replace $K(x)=1/x$ with

$K_{i}(x)={\frac {x_{i}}{|x|^{n+1}}}$

where $i=1,...,n$ and $x_{i}$ is the $i$ -th component of $x$ in $\mathbb {R} ^{n}$ . All of these operators are bounded on $L^{p}$ and satisfy weak-type $(1,1)$ estimates.

02Singular integrals of convolution type

A singular integral of convolution type is an operator $T$ defined by convolution with a kernel $K$ that is locally integrable on $R^{n}\setminus \{0\}$ , in the sense that

$T(f)(x)=\lim _{\varepsilon \to 0}\int _{|y-x|>\varepsilon }K(x-y)f(y)\,dy.$

Suppose that the kernel satisfies:

The size condition on the Fourier transform of $K$
${\hat {K}}\in L^{\infty }(\mathbb {R} ^{n})$
The smoothness condition: for some $C>0$ ,
$\sup _{y\neq 0}\int _{|x|>2|y|}|K(x-y)-K(x)|\,dx\leq C.$

Then it can be shown that $T$ is bounded on $L^{p}(\mathbb {R} ^{n})$ and satisfies a weak-type $(1,1)$ estimate.

Property 1. is needed to ensure that convolution (1) with the tempered distribution p.v. $K$ given by the principal value integral

$\operatorname {p.v.} \,\,K[\phi ]=\lim _{\epsilon \to 0^{+}}\int _{|x|>\epsilon }\phi (x)K(x)\,dx$

is a well-defined Fourier multiplier on $L^{2}$ . Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist. Typically in applications, one also has a cancellation condition

$\int _{R_{1}<|x|<R_{2}}K(x)\,dx=0,\ \forall R_{1},R_{2}>0$

which is quite easy to check. It is automatic, for instance, if $K$ is an odd function. If, in addition, one assumes 2. and the following size condition

$\sup _{R>0}\int _{R<|x|<2R}|K(x)|\,dx\leq C,$

then it can be shown that 1. follows.

The smoothness condition 2. is also often difficult to check in principle, the following sufficient condition of a kernel $K$ can be used:

$K\in C^{1}(\mathbf {R} ^{n}\setminus \{0\})$
$|\nabla K(x)|\leq {\frac {C}{|x|^{n+1}}}$

Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.

03Singular integrals of non-convolution type

These are even more general operators. However, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on $L^{p}$ .

Calderón-Zygmund kernels

A function $K:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R}$ is said to be a Calderón, Zygmund kernel if it satisfies the following conditions for some constants $C>0$ and $\delta >0$ .

$|K(x,y)|\leq {\frac {C}{|x-y|^{n}}}$
$|K(x,y)-K(x',y)|\leq {\frac {C|x-x'|^{\delta }}{{\bigl (}|x-y|+|x'-y|{\bigr )}^{n+\delta }}}{\text{ whenever }}|x-x'|\leq {\frac {1}{2}}\max {\bigl (}|x-y|,|x'-y|{\bigr )}$
$|K(x,y)-K(x,y')|\leq {\frac {C|y-y'|^{\delta }}{{\bigl (}|x-y|+|x-y'|{\bigr )}^{n+\delta }}}{\text{ whenever }}|y-y'|\leq {\frac {1}{2}}\max {\bigl (}|x-y'|,|x-y|{\bigr )}$

Singular integrals of non-convolution type

$T$ is said to be a singular integral operator of non-convolution type associated to the Calderón, Zygmund kernel $K$ if

$\int g(x)T(f)(x)\,dx=\iint g(x)K(x,y)f(y)\,dy\,dx,$

whenever $f$ and $g$ are smooth and have disjoint support. Such operators need not be bounded on $L^{p}$

Calderón, Zygmund operators

A singular integral of non-convolution type $T$ associated to a Calderón, Zygmund kernel $K$ is called a Calderón, Zygmund operator when it is bounded on $L^{2}$ , that is, there is a $C>0$ such that

$\|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}},$

for all smooth compactly supported ƒ.

It can be proved that such operators are, in fact, also bounded on all $L^{p}$ with $1<p<\infty$ .

The $T(b)$ theorem

The $T(b)$ theorem provides sufficient conditions for a singular integral operator to be a Calderón-Zygmund operator, that is for a singular integral operator associated to a Calderón-Zygmund kernel to be bounded on $L^{2}$ . In order to state the result we must first define some terms.

A normalised bump is a smooth function $\varphi$ on $\mathbb {R} ^{n}$ supported in a ball of radius 1 and centred at the origin such that $|\partial ^{\alpha }\varphi (x)|<1$ , for all multi-indices $|\alpha |\leq n+2$ . Denote by $\tau ^{x}(\varphi )(y)=\varphi (y-x)$ and $\varphi _{r}(x)=r^{-n}\varphi (x/r)$ for all $x$ in $\mathbb {R} ^{n}$ and $r>0$ . An operator is said to be weakly bounded if there is a constant $C$ such that

$\left|\int T{\bigl (}\tau ^{x}(\varphi _{r}){\bigr )}(y)\tau ^{x}(\psi _{r})(y)\,dy\right|\leq Cr^{-n}$

for all normalised bumps $\varphi$ and $\psi$ . A function is said to be accretive if there is a constant $c>0$ such that $Re(b)(x)\geq c$ for all $x$ in $\mathbb {R}$ . Denote by $M_{b}$ the operator given by multiplication by a function $b$ .

The $T(b)$ theorem states that a singular integral operator $T$ associated to a Calderón-Zygmund kernel is bounded on $L^{2}$ if it satisfies all of the following three conditions for some bounded accretive functions $b_{1}$ and $b_{2}$ :

$M_{b_{2}}TM_{b_{1}}$ is weakly bounded;
$T(b_{1})$ is in BMO;
$T^{t}(b_{2}),$ is in BMO, where $T^{t}$ is the transpose operator of $T$ .

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Singular integral

01The Hilbert transform

02Singular integrals of convolution type