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# Singular integrals

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### Singular integrals

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

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

whose kernel function K : Rn×Rn → Rn is singular along the diagonal x = y. Specifically, the singularity is such that |K(xy)| 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 Lp(Rn).

## The Hilbert transform

Main article: 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,

$H\left(f\right)\left(x\right) = \frac\left\{1\right\}\left\{\pi\right\}\lim_\left\{\varepsilon \to 0\right\} \int_\left\{|x-y|>\varepsilon\right\} \frac\left\{1\right\}\left\{x-y\right\}f\left(y\right) \, dy.$

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

$K_i\left(x\right) = \frac\left\{x_i\right\}\left\{|x|^\left\{n+1\right\}\right\}$

where i = 1, …, n and $x_i$ is the i-th component of x in Rn. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.[1]

## Singular integrals of convolution type

Main article: Singular integral operators 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 Rn\{0}, in the sense that

y-x

(>\varepsilon} K(x-y)f(y) \, dy. )

Suppose that the kernel satisfies:

1. The size condition on the Fourier transform of K

$\hat\left\{K\right\}\in L^\infty\left(\mathbf\left\{R\right\}^n\right)$

2. The smoothness condition: for some C > 0,

$\sup_\left\{y \neq 0\right\} \int_\left\{|x|>2|y|\right\} |K\left(x-y\right) - K\left(x\right)| \, dx \leq C.$

Then it can be shown that T is bounded on Lp(Rn) 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

$\operatorname\left\{p.v.\right\}\,\, K\left[\phi\right] = \lim_\left\{\epsilon\to 0^+\right\} \int_\left\{|x|>\epsilon\right\}\phi\left(x\right)K\left(x\right)\,dx$

is a well-defined Fourier multiplier on L2. 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

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

$\sup_\left\{R>0\right\} \int_\left\{R<|x|<2R\right\} |K\left(x\right)| \, dx \leq C,$

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:

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

Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.[2]

## 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 : Rn×Rn → R is said to be a CalderónZygmund kernel if it satisfies the following conditions for some constants C > 0 and δ > 0.[2]

$\left(a\right) \qquad |K\left(x,y\right)| \leq \frac\left\{C\right\}\left\{|x-y|^n\right\}$
$\left(b\right) \qquad |K\left(x,y\right) - K\left(x\text{'},y\right)| \leq \frac\left\{C|x-x\text{'}|^\delta\right\}\left\{\left(|x-y|+|x\text{'}-y|\right)^\left\{n+\delta\right\}\right\}\text\left\{ whenever \right\}|x-x\text{'}| \leq \frac\left\{1\right\}\left\{2\right\}\max\left(|x-y|,|x\text{'}-y|\right)$
$\left(c\right) \qquad |K\left(x,y\right) - K\left(x,y\text{'}\right)| \leq \frac\left\{C|y-y\text{'}|^\delta\right\}\left\{\left(|x-y|+|x-y\text{'}|\right)^\left\{n+\delta\right\}\right\}\text\left\{ whenever \right\}|y-y\text{'}| \leq \frac\left\{1\right\}\left\{2\right\}\max\left(|x-y\text{'}|,|x-y|\right)$

### 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

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

whenever f and g are smooth and have disjoint support.[2] Such operators need not be bounded on Lp

### 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 L2, that is, there is a C > 0 such that

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

for all smooth compactly supported ƒ.

It can be proved that such operators are, in fact, also bounded on all Lp 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 L2. In order to state the result we must first define some terms.

A normalised bump is a smooth function φ on Rn supported in a ball of radius 10 and centred at the origin such that |∂α φ(x)| ≤ 1, for all multi-indices |α| ≤ n + 2. Denote by τx(φ)(y) = φ(y − x) and φr(x) = r−nφ(x/r) for all x in Rn and r > 0. An operator is said to be weakly bounded if there is a constant C such that

$\left|\int T\left(\tau^x\left(\varphi_r\right)\right)\left(y\right) \tau^x\left(\psi_r\right)\left(y\right) \, dy\right| \leq Cr^\left\{-n\right\}$

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 Mb 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 L2 if it satisfies all of the following three conditions for some bounded accretive functions b1 and b2:[3]

(a) $M_\left\{b_2\right\}TM_\left\{b_1\right\}$ is weakly bounded;

(b) $T\left(b_1\right)$ is in BMO;

(c) $T^t\left(b_2\right),$ is in BMO, where Tt is the transpose operator of T.

## References

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