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Concept
Singular integral
Summary
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
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 Lp(Rn).
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,
The most straightforward higher dimension analogues of these are the Riesz transforms, which replace K(x) = 1/x with
where i = 1, ..., n and is the i-th component of x in Rn. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.
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
Suppose that the kernel satisfies:
The size condition on the Fourier transform of K
The smoothness condition: for some C > 0,
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 () with the tempered distribution p.v. K given by the principal value integral
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
then it can be shown that 1.
Official source
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