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Wikipedia / 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

is singular, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp spaces.

The Hilbert transform

The archetypal singular integral operator is the Hilbert transform. More precisely,

The most straightforward higher dimension analogues of these are the Riesz transforms with

and satisfy weak-type estimates.

Singular integrals of convolution type

that is locally integrable, in the sense that

Suppose that the kernel satisfies:

  1. The size condition on the Fourier transform

  2. The smoothness,

estimate.

Property 1. is needed to ensure that convolution () with the tempered distribution p.v. given by the principal value integral

is a well-defined Fourier multiplier. 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

is an odd function. If, in addition, one assumes 2. and the following size condition

then it can be shown that 1. follows.

can be used:

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

.

Calderón–Zygmund kernels

is said to be a Calderón–Zygmund kernel.

Singular integrals of non-convolution type

is said to be a singular integral operator of non-convolution type associated to the Calderón if

are smooth and have disjoint support.

Calderón–Zygmund operators

associated to a Calderón is called a CalderónZygmund operator such that

for all smooth compactly supported ƒ.

.

T(b) theorem

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

A normalised bump. An operator is said to be weakly bounded such that

. A function is said to be accretive.

: is weakly bounded; is in BMO; is in BMO is the transpose operator of.

See also

Notes

References

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  • (in Russian).

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  • , (European edition: ).