Singular integral

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" : R^"n"×R^"n" → R^"n" 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"(R^"n").

The Hilbert transform

The archetypal singular integral operator is the Hilbert transform "H". It is given by convolution against the kernel "K"("x") = 1/(π"x") for "x" in R. More precisely,

: $H(f)(x)\; =\; frac\{1\}\{pi\}lim\_\{varepsilon\; o\; 0\}\; int\_\; |K(x-y)\; -\; K(x)|\; ,\; dx\; leq\; C$

and the "cancellation" condition

:

Then we know that "T" is bounded on "L"^p(R^"n") and satisfies a weak-type (1,1) estimate. Observe that these conditions are satisfied for the Hilbert and Reisz transforms, so this result is an extension of those result. cite book | last = Grakakos | first = Loukas | coauthors = | title = Classical and Modern Fourier Analysis | chapter = 7 | work = | pages = | language = | publisher = Pearson Education, Inc. | place = New Jersey| date = 2004 | url = | accessdate = ]

ingular 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" : R^"n"×R^"n" → R is said to be "Calderón-Zygmund kernel" if it satisfies the following conditions for some constants "C" > 0 and δ > 0.

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

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

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

ingular 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

: $int\; g(x)\; T(f)(x)\; ,\; dx\; =\; iint\; g(x)\; K(x-y)\; f(y)\; ,\; dy\; ,\; dx,$

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 "L"², 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 "L"². In order to state the result we must first define some terms.

A "normalised bump" is a smooth function φ on R^"n" 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 R^"n" and "r" > 0. An operator is said to be "weakly bounded" if there is a constant "C" such that

: $|int\; T(\; au^x(varphi\_r))(y)\; au^x(psi\_r)(y)\; ,\; dy|\; leq\; Cr^\{-n\}$

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 "L"² if it satisfies all of the following three conditions for some bounded accretive functions "b"₁ and "b"₂: [cite article | last = David | first = | coauthors = Journé | coauthors = Semmes | title = Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation | publisher = Revista Matemática Iberoamericana | volume = 1 | pages = 1&ndash;56| language = French | date = 1985 | url = | accessdate = ]

(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".

Notes