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User:Potahto/Singular integrals

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Singular integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking they are operators of order zero which arise from kernels via the expression

where is of size , and so the kernels are singular along . Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over as , but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on .

The Hilbert transform

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The archetypal singular integral operator is the Hilbert transform . It is given by convolution against the kernel More precisely,

The most straightforward higher dimension analogues of these are the Reisz transforms, which replace with

where is the th component of . All of these operators are bounded on and satisfy weak-type estimates.[1]

Singular integrals of convolution type

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A singular integral of convolution type is an operator defined by convolution again a kernel in the sense that

Suppose that, for some , the kernel satisfies the size condition

the smoothness condition

and the cancellation condition

Then we know that is bounded on and satisfies a weak-type estimate. Observe that these conditions are satisfies for the Hilbert and Reisz transforms, so this result is an extension of those result.[2]

Singular integrals of non-convolution type

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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 .

Calderón-Zygmund kernels

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A function is said to be Calderón-Zygmund kernel if it satisfies the following conditions for some constants and .[2]

(a)

(b) whenever

(c) whenever


Singular Integrals of non-convolution type

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A singular integral of non-convolution type is an operator associated to a Calderón-Zygmund kernel is an operator which is such that

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

Calderón-Zygmund operators

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A singular integral of non-convolution type associated to a Calderón-Zygmund kernel is called a Calderón-Zygmund operator when it is bounded on , that is, there is a such that

for all smooth compactly supported .

It can be proved that such operators are, in fact, also bounded on all for .

The T(b) Theorem

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The 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 . In order to state the result we must first define some terms.

A normalised bump is a smoth function on supported in a ball of radius 10 and centred at the origin such that , for all multi-indices . Denote by and for and . An operator is said to be weakly bounded if there is a constant such that

for all normalised bumps and . A function is said to be coercive if there is a constant such that for all . Denote by the operator given by multiplication by a function .

The Theorem states that a singular integral operator associated to a Calderón-Zygmund kernel is bounded on if it satisfies all of the following three condtions for some bounded accretive functions and :[3]

(a) is weakly bounded;

(b)

(c) ,where is the transpose operator of .

Notes

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  1. ^ Stein, Elias (1993). "Harmonic Analysis". Princeton University Press. {{cite news}}: Check date values in: |date= (help); Cite has empty unknown parameter: |coauthors= (help)
  2. ^ a b c Grakakos, Loukas (2004). "7". Classical and Modern Fourier Analysis. New Jersey: Pearson Education, Inc. {{cite book}}: Check date values in: |date= (help); Cite has empty unknown parameter: |coauthors= (help)
  3. ^ David; Journé; Semmes (1985). "Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation" (in French). Vol. 1. Revista Matemática Iberoamericana. pp. 1–56. {{cite news}}: Check date values in: |date= (help)