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 : Rⁿ×Rⁿ → 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(Rⁿ).

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 Rⁿ. All of these operators are bounded on L^p and satisfy weak-type (1, 1) estimates.{{cite news | last = Stein | first = Elias | title = Harmonic Analysis | publisher = Princeton University Press| year = 1993 }}

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 Rⁿ\{0}, in the sense that

{{NumBlk|:| $T(f)(x) = \lim_{\varepsilon \to 0} \int_$

y-x|>\varepsilon} K(x-y)f(y) \, dy. |{{EquationRef|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(Rⁿ) and satisfies a weak-type (1, 1) estimate.

Property 1. is needed to ensure that convolution ({{EquationNote|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 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

: $\int_{R_1<|x| 0$

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:

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

Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.{{Citation | last = Grafakos | first = Loukas | title = Classical and Modern Fourier Analysis | chapter = 7 | publisher = Pearson Education, Inc. | place = New Jersey| year = 2004 }}

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 {{nowrap|K : Rⁿ×Rⁿ → 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 L², 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 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 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⁻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

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

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 L² if it satisfies all of the following three conditions for some bounded accretive functions b₁ and b₂:{{cite news | last = David |author3=Journé |author2=Semmes | title = Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation | publisher = Revista Matemática Iberoamericana | volume = 1 | pages = 1–56| language = fr | year = 1985 }}

$M_{b_2}TM_{b_1}$ 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.

Notes

References

{{Citation

| last1=Calderon

| first1=A. P.

| author-link = Alberto Calderón

| last2=Zygmund

| first2=A.

| author2-link = Antoni Zygmund

| title=On the existence of certain singular integrals

| issue = 1

| mr= 0052553

| zbl = 0047.10201

| year=1952

| journal=Acta Mathematica

| doi = 10.1007/BF02392130

| issn=0001-5962

| volume=88

| pages=85–139| doi-access=free

}}.

{{Citation

| doi=10.2307/2372517

| last1=Calderon

| first1=A. P.

| author-link = Alberto Calderón

| last2=Zygmund

| first2=A.

| author2-link = Antoni Zygmund

| title=On singular integrals

| mr=0084633 | zbl = 0072.11501

| year=1956

| journal=American Journal of Mathematics

| issue=2

| issn=0002-9327

| volume=78

| pages=289–309

| publisher=The Johns Hopkins University Press

| jstor=2372517}}.

{{citation

| last2=Meyer

| first2=Yves

| author-link2=Yves Meyer

| last1=Coifman

| first1= Ronald

| author-link1=Ronald Coifman

| title=Wavelets: Calderón-Zygmund and multilinear operators

| series= Cambridge Studies in Advanced Mathematics

| volume= 48

| publisher= Cambridge University Press

| year= 1997

| pages=xx+315

| isbn= 0-521-42001-6

| mr=1456993

| zbl=0916.42023

}}.

{{Citation

| last = Mikhlin

| first = Solomon G.

| author-link = Solomon Mikhlin

| title = Singular integral equations

| journal = UMN

| volume = 3

| issue = 25

| pages = 29–112

| year = 1948

| url = http://mi.mathnet.ru/eng/umn/v3/i3/p29

| mr = 27429

}} (in Russian).

{{Citation

| last = Mikhlin

| first = Solomon G.

| author-link = Solomon Mikhlin

| title = Multidimensional singular integrals and integral equations

| place = Oxford–London–Edinburgh–New York City–Paris–Frankfurt

| publisher = Pergamon Press

| year = 1965

| series = International Series of Monographs in Pure and Applied Mathematics

| volume = 83

| mr=0185399

| zbl = 0129.07701

| pages = XII+255

}}.

{{Citation

| last1 = Mikhlin

| first1 = Solomon G.

| author-link = Solomon Mikhlin

| last2 = Prössdorf

| first2 = Siegfried

| title = Singular Integral Operators

| place = Berlin–Heidelberg–New York City

| publisher = Springer Verlag

| year = 1986

| pages = 528

| url = https://books.google.com/books?id=eaMmy99UTHgC

| mr=0867687

| zbl=0612.47024

| isbn = 0-387-15967-3}}, (European edition: {{ISBN|3-540-15967-3}}).

{{Citation

| first=Elias

| last=Stein

| author-link=Elias Stein

| title=Singular integrals and differentiability properties of functions

| series= Princeton Mathematical Series

| volume=30

| publisher=Princeton University Press

| location = Princeton, NJ

| url = https://books.google.com/books?id=sAWpsmkqziEC&q=Singular+integrals+and+differentiability+properties+of+functions

| year=1970

| pages = XIV+287

| isbn=0-691-08079-8

| mr=0290095

| zbl = 0207.13501}}

External links

{{cite journal | last = Stein | first = Elias M. |date=October 1998 | title = Singular Integrals: The Roles of Calderón and Zygmund | journal = Notices of the American Mathematical Society | volume = 45 | issue = 9 | pages = 1130–1140 | url = http://www.ams.org/notices/199809/stein.pdf }}

Category:Harmonic analysis

Category:Real analysis

Singular integral

The Hilbert transform

Singular integrals of convolution type

Singular integrals of non-convolution type

=Calderón–Zygmund kernels=

=Singular integrals of non-convolution type=

=Calderón&ndash;Zygmund operators=

=The ''T''(''b'') theorem=

See also

Notes

References

External links

=Calderón–Zygmund operators=