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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 ) = 1 π lim ε → 0 ∫ | x − y | > ε 1 x − y f ( y ) d y . {\displaystyle 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 ) = x i | x | n + 1 {\displaystyle K_{i}(x)={\frac {x_{i}}{|x|^{n+1}}}}

where i = 1, ..., n and x i {\displaystyle x_{i}} is the i-th component of x in Rn. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.¹

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 ε → 0 ∫ | y − x | > ε K ( x − y ) f ( y ) d y . {\displaystyle 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 K ^ ∈ L ∞ ( R n ) {\displaystyle {\hat {K}}\in L^{\infty }(\mathbf {R} ^{n})}
2. The smoothness condition: for some C > 0, sup y ≠ 0 ∫ | x | > 2 | y | | K ( x − y ) − K ( x ) | d x ≤ C . {\displaystyle \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 Lp(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

p . v . K [ ϕ ] = lim ϵ → 0 + ∫ | x | > ϵ ϕ ( x ) K ( x ) d x {\displaystyle \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

∫ R 1 < | x | < R 2 K ( x ) d x = 0 ,   ∀ R 1 , R 2 > 0 {\displaystyle \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 ∫ R < | x | < 2 R | K ( x ) | d x ≤ C , {\displaystyle \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 ∈ C 1 ( R n ∖ { 0 } ) {\displaystyle K\in C^{1}(\mathbf {R} ^{n}\setminus \{0\})}
• | ∇ K ( x ) | ≤ C | x | n + 1 {\displaystyle |\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.²

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.

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 > 0.³

1. | K ( x , y ) | ≤ C | x − y | n {\displaystyle |K(x,y)|\leq {\frac {C}{|x-y|^{n}}}}
2. | K ( x , y ) − K ( x ′ , y ) | ≤ C | x − x ′ | δ ( | x − y | + | x ′ − y | ) n + δ  whenever  | x − x ′ | ≤ 1 2 max ( | x − y | , | x ′ − y | ) {\displaystyle |K(x,y)-K(x',y)|\leq {\frac {C|x-x'|^{\delta }}{{\bigl (}|x-y|+|x'-y|{\bigr )}^{n+\delta }}}{\text{ whenever }}|x-x'|\leq {\frac {1}{2}}\max {\bigl (}|x-y|,|x'-y|{\bigr )}}
3. | K ( x , y ) − K ( x , y ′ ) | ≤ C | y − y ′ | δ ( | x − y | + | x − y ′ | ) n + δ  whenever  | y − y ′ | ≤ 1 2 max ( | x − y ′ | , | x − y | ) {\displaystyle |K(x,y)-K(x,y')|\leq {\frac {C|y-y'|^{\delta }}{{\bigl (}|x-y|+|x-y'|{\bigr )}^{n+\delta }}}{\text{ whenever }}|y-y'|\leq {\frac {1}{2}}\max {\bigl (}|x-y'|,|x-y|{\bigr )}}

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

∫ g ( x ) T ( f ) ( x ) d x = ∬ g ( x ) K ( x , y ) f ( y ) d y d x , {\displaystyle \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.⁴ Such operators need not be bounded on Lp

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 ≤ C ‖ f ‖ L 2 , {\displaystyle \|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 Lp 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 τx(φ)(y) = φ(y − x) and φr(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

| ∫ T ( τ x ( φ r ) ) ( y ) τ x ( ψ r ) ( y ) d y | ≤ C r − n {\displaystyle \left|\int T{\bigl (}\tau ^{x}(\varphi _{r}){\bigr )}(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 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 conditions for some bounded accretive functions b1 and b2:⁵

1. M b 2 T M b 1 {\displaystyle M_{b_{2}}TM_{b_{1}}} is weakly bounded;
2. T ( b 1 ) {\displaystyle T(b_{1})} is in BMO;
3. T t ( b 2 ) , {\displaystyle T^{t}(b_{2}),} is in BMO, where Tt is the transpose operator of T.

See also

• Singular integral operators on closed curves

Notes

• Calderon, A. P.; Zygmund, A. (1952), "On the existence of certain singular integrals", Acta Mathematica, 88 (1): 85–139, doi:10.1007/BF02392130, ISSN 0001-5962, MR 0052553, Zbl 0047.10201.
• Calderon, A. P.; Zygmund, A. (1956), "On singular integrals", American Journal of Mathematics, 78 (2), The Johns Hopkins University Press: 289–309, doi:10.2307/2372517, ISSN 0002-9327, JSTOR 2372517, MR 0084633, Zbl 0072.11501.
• Coifman, Ronald; Meyer, Yves (1997), Wavelets: Calderón-Zygmund and multilinear operators, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, pp. xx+315, ISBN 0-521-42001-6, MR 1456993, Zbl 0916.42023.
• Mikhlin, Solomon G. (1948), "Singular integral equations", UMN, 3 (25): 29–112, MR 0027429 (in Russian).
• Mikhlin, Solomon G. (1965), Multidimensional singular integrals and integral equations, International Series of Monographs in Pure and Applied Mathematics, vol. 83, Oxford–London–Edinburgh–New York City–Paris–Frankfurt: Pergamon Press, pp. XII+255, MR 0185399, Zbl 0129.07701.
• Mikhlin, Solomon G.; Prössdorf, Siegfried (1986), Singular Integral Operators, Berlin–Heidelberg–New York City: Springer Verlag, p. 528, ISBN 0-387-15967-3, MR 0867687, Zbl 0612.47024, (European edition: ISBN 3-540-15967-3).
• Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, Princeton, NJ: Princeton University Press, pp. XIV+287, ISBN 0-691-08079-8, MR 0290095, Zbl 0207.13501

External links

• Stein, Elias M. (October 1998). "Singular Integrals: The Roles of Calderón and Zygmund" (PDF). Notices of the American Mathematical Society. 45 (9): 1130–1140.

References

1. Stein, Elias (1993). "Harmonic Analysis". Princeton University Press. ↩

2. Grafakos, Loukas (2004), "7", Classical and Modern Fourier Analysis, New Jersey: Pearson Education, Inc. ↩

3. Grafakos, Loukas (2004), "7", Classical and Modern Fourier Analysis, New Jersey: Pearson Education, Inc. ↩

4. Grafakos, Loukas (2004), "7", Classical and Modern Fourier Analysis, New Jersey: Pearson Education, Inc. ↩

5. David; Semmes; Journé (1985). "Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation" (in French). Vol. 1. Revista Matemática Iberoamericana. pp. 1–56. ↩