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