Singular integral

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In mathematics, singular integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular intetgral is an integral operator

T(f)(x) = \int K(x,y)f(y) \, dy,

whose kernel function K : Rn×Rn → Rn 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| > \varepsilon as \varepsilon \to 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on L^p(\mathbb{R}^n).

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[edit] The Hilbert transform

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 / x (x \in \mathbb{R}). More precisely,

H(f)(x) = \lim_{\varepsilon \to 0} \int_{|y-x|>\varepsilon} \frac{1}{y-x}f(y) \, dy.

The most straightforward higher dimension analogues of these are the Reisz transforms, which replace K(x) = 1 / x with

K_i(x) = \frac{x_i}{|x|^2}

(i = 1,\dots,n), where xi is the ith component of x \in \mathbb{R}^n. All of these operators are bounded on Lp and satisfy weak-type (1,1) estimates.[1]

[edit] Singular integrals of convolution type

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

T(f)(x) = \lim_{\varepsilon \to 0} \int_{|y-x|>\varepsilon} K(x-y)f(y) \, dy.

Suppose that, for some C > 0, the kernel satisfies the size condition

\sup_{R>0} \int_{R<|x|<2R} |K(x)| \, dx \leq C,

the smoothness condition

\sup_{y \neq 0} \int_{|x|<2|y|} |K(x-y) - K(x)| \, dx \leq C

and the cancellation condition

\sup_{0<R_1,R_2<\infty} |\int_{R_1<|x|<R_2} K(x) \, dx| \leq C.

Then we know that T is bounded on L^p(\mathbb{R}^n) and satisfies a weak-type (1,1) estimate. Observe that these conditions are satisfies for the Hilbert and Reisz transforms, so this result is an extension of those result.[2]

[edit] 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 Lp.

[edit] Calderón-Zygmund kernels

A function K \colon \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} is said to be Calderón-Zygmund kernel if it satisfies the following conditions for some constants C > 0 and δ > 0.[2]

(a) |K(x,y)| \leq \frac{C}{|x-y|}

(b) |K(x,y) - K(x',y)| \leq \frac{C|x-x'|^\delta}{(|x-y|+|x'-y|)^{n+\delta}} whenever |x-x'| \leq \frac{1}{2}\max(|x-y|,|x'-y|)

(c) |K(x,y) - K(x,y')| \leq \frac{C|y-y'|^\delta}{(|x-y|+|x'-y|)^{n+\delta}} whenever |y-y'| \leq \frac{1}{2}\max(|x-y'|,|x-y|)

[edit] Singular Integrals of non-convolution type

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

\int g(x) T(f)(x) \, dx = \iint g(x) K(x-y) f(y) \, dydx,

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

[edit] 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 f.

It can be proved that such operators are, in fact, also bounded on all Lp for p \in (1,\infty).

[edit] 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 \mathbb{R}^n supported in a ball of radius 10 and centred at the origin such that |\partial^\alpha \phi(x)| \leq 1, for all multi-indices |\alpha| \leq n + 2. Denote by τx(φ)(y) = φ(y − x) and φr(x) = r − nφ(x / r) for x \in \mathbb{R}^n and r > 0. An operator is said to be weakly bounded if there is a constant C such that

|\int T(\tau^x(\phi_r))(y) \tau^x(\psi_r)(y) \, dy| \leq Cr^{-n}

for all normalised bumps φ and ψ. A function is said to be coercive if there is a constant c > 0 such that \Re(b)(x) \geq c for all x \in \mathbb{R}. Denote by Mb 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 condtions for some bounded accretive functions b1 and b2:[3]

(a) M_{b_2}TM_{b_1} is weakly bounded;

(b) T(b1) = 0;

(c) Tt(b2) = 0,where Tt is the transpose operator of T.

[edit] Notes

  1. ^ Stein, Elias. "Harmonic Analysis", Princeton University Press, 1993. 
  2. ^ a b c Grakakos, Loukas (2004). "7", Classical and Modern Fourier Analysis. Pearson Education, Inc.. 
  3. ^ David; Semmes. "Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation", Revista Matemática Iberoamericana, 1985, pp. 1-56. (French)