Singular integral explained

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

See 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(f)(x) = (1)/(\pi) \lim _{\varepsilon} \int _{|x-y|>\varepsilon} (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)} = (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.^[1]

Singular integrals of convolution type

See 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\, in the sense that

Suppose that the kernel satisfies:

1. The size condition on the Fourier transform of K

\hat{K}\in L ^{infty(R} ^{n)}

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

\sup _{y} \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 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|<R} _{2}} K(x) dx = 0 , \forall R _{1,R} _{2} > 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(R} ^{n\setminus\{0\})} |\nabla K(x)|\le (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.

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 is said to be a Calderón–Zygmund kernel if it satisfies the following conditions for some constants C > 0 and δ > 0.

1. |K(x,y)| \leq (C)/(|x - y| ^{n})
2. |K(x,y) - K(x',y)| \leq (C|x - x'| ^{\delta})/(l (|x - y| + |x' - y| r ) ^{n+\delta}) whenever |x - x'| \leq (1)/(2) max l (|x - y|,|x' - y| r )
3. |K(x,y) - K(x,y')| \leq (C |y - y'| ^{\delta})/(l (|x - y| + |x - y'| r ) ^{n+\delta}) whenever |y - y'| \leq (1)/(2) max l (|x - y'|,|x - y| r )

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 l (\tau ^{x(\varphi} _{r) r )(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:^[2]

1. M _{b _{2}} TM _{b _{1}} is weakly bounded;
2. T(b _{1)} is in BMO;
3. T ^{t(b} _{2 ),} is in BMO, where T^t is the transpose operator of T.

See also

• Singular integral operators on closed curves

Notes

1. News: Stein . Elias . Harmonic Analysis . Princeton University Press. 1993 .
2. News: David . Journé . Semmes . Opérateurs de Calderón - Zygmund, fonctions para-accrétives et interpolation . Revista Matemática Iberoamericana . 1 . 1 - 56. fr . 1985 .

References

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• (in Russian).
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• , (European edition:).

External links

• Stein . Elias M. . October 1998 . Singular Integrals: The Roles of Calderón and Zygmund . . 45 . 9 . 1130–1140 .