# Calderón-Zygmund operator

An operator defined on a space of sufficiently smooth functions with compact support in by the formula

where the kernel is a homogeneous function of degree with zero mean value over the unit sphere . The kernel has the form

where , the characteristic function of , satisfies the conditions

(*) |

The Calderón–Zygmund operator is usually written in the form

here denotes the principal value of the integral. In the one-dimensional case the Calderón–Zygmund operator becomes the Hilbert operator :

The Calderón–Zygmund operator can be extended by continuity to the space of functions in that are summable of degree . This extension maps continuously into itself. If satisfies the condition (*) and also Dini's condition:

and if

for and , then

a) there exists a constant (independent of or ) such that

b) the limit exists in the sense of convergence in and

The Calderón–Zygmund operator was analyzed by A.P. Calderón and A. Zygmund [1].

#### References

[1] | A.P. Calderón, A. Zygmund, "On the existence of certain singular integrals" Acta Math. , 88 (1952) pp. 85–139 |

[2] | S.G. Mikhlin, "Multidimensional singular integrals and integral equations" , Pergamon (1965) (Translated from Russian) |

[3] | E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970) |

#### Comments

Proofs of the estimates a) and b) above can be found in [3], Chapt. II, Section 4.

In the 1960's the operators described above were often called Mikhlin–Calderón–Zygmund operators, since the main boundedness theorem for them on (estimate a)) was proved by S.G. Mikhlin in 1938 (published in [a1]).

Nowadays the term Calderón–Zygmund operator is often used for general operators in the class defined by G. David and J.L. Journé in [a2].

Assertion b) has been generalized to singular integrals with odd kernels, and to singular integrals with even kernels , [a3], Chapt. VI, Sections 2, 3.

See also Singular integral; Hilbert singular integral; Hilbert transform.

#### References

[a1] | S.G. Mikhlin, "On a boundedness theorem for a singular integral operator" Uspekhi Mat. Nauk , 8 (1953) pp. 213–217 (In Russian) |

[a2] | G. David, J.L. Journé, "Une characterization des opérateurs intégraux singuliers bornés sur " C.R. Acad. Sci. Paris , 296 (1983) pp. 761–764 |

[a3] | E.M. Stein, G. Weiss, "Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1975) |

**How to Cite This Entry:**

Calderón–Zygmund operator.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Calder%C3%B3n%E2%80%93Zygmund_operator&oldid=38685