Singular integral
# Singular integral

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 : $R^n×R^n$ → 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| → 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| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on $L^p(R^n).$

## The Hilbert transform

The archetypal singular integral operator is the Hilbert transform H. It is given by convolution against the kernel K(x) = 1/(πx) for x in 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 $R^n.$ All of these operators are bounded on $L^p$ and satisfy weak-type (1, 1) estimates.

## Singular 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 $R^n\{0},$ in the sense that

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

Suppose that the kernel satisfies:

* The size condition on the Fourier transform of K

    ${\hat {K}}\in L^{\infty }(\mathbf {R} ^{n})$
* 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 $L^p(R^n)$ 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

    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \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.

## Singular 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 : $R^n×R^n$ → R is said to be a Calderón–Zygmund kernel if it satisfies the following conditions for some constants C > 0 and δ > 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 < ∞.

### 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 φ on $R^n$ supported in a ball of radius 1 and centred at the origin such that $|∂^α$ φ(x)| ≤ 1, for all multi-indices |α| ≤ n + 2. Denote by $τ^x(φ)(y)$ = φ(y − x) and $φ_r(x)$ = $r^{−n}φ(x/r)$ for all x in $R^n$ 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 φ and ψ. A function is said to be accretive if there is a constant c > 0 such that Re(b)(x) ≥ c for all x in 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.
