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

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

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| > 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

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,

$H(f)(x)={\frac {1}{\pi }}\lim _{\varepsilon \to 0}\int _{|x-y|>\varepsilon }{\frac {1}{x-y}}f(y)\,dy.$

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

$K_{i}(x)={\frac {x_{i}}{|x|^{n+1}}}$

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.

Singular integrals of convolution type

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\{0}, in the sense that

$T(f)(x)=\lim _{\varepsilon \to 0}\int _{|y-x|>\varepsilon }K(x-y)f(y)\,dy.$ (1)

Suppose that the kernel satisfies:

1. The size condition on the Fourier transform of K

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

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

$\sup _{y\neq 0}\int _{|x|>2|y|}|K(x-y)-K(x)|\,dx\leq C.$

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 (1) 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:

• $K\in C^{1}(\mathbf {R} ^{n}\setminus \{0\})$
• $|\nabla K(x)|\leq {\frac {C}{|x|^{n+1}}}$

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

$(a)\qquad |K(x,y)|\leq {\frac {C}{|x-y|^{n}}}$

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

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

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 10 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

$\left|\int T(\tau ^{x}(\varphi _{r}))(y)\tau ^{x}(\psi _{r})(y)\,dy\right|\leq Cr^{-n}$

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:

(a) $M_{b_{2}}TM_{b_{1}}$ is weakly bounded;

(b) $T(b_{1})$ is in BMO;

(c) $T^{t}(b_{2}),$ is in BMO, where T^t is the transpose operator of T.

References

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