# Calderón-Zygmund operator

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An operator $K$ defined on a space of sufficiently smooth functions $\phi$ with compact support in $\mathbf R ^ {n}$ by the formula

$$K \phi (x) = \ \lim\limits _ {\epsilon \rightarrow 0 } \ \int\limits _ {| x - y | > \epsilon } k (x - y) \phi (y) dy,$$

where the kernel $k$ is a homogeneous function of degree $-n$ with zero mean value over the unit sphere $S ^ {n - 1 } = \{ {x } : {x \in \mathbf R ^ {n} , | x | = 1 } \}$. The kernel $k$ has the form

$$k (x) = \ \frac{\Omega (x) }{| x | ^ {n} } ,$$

where $\Omega$, the characteristic function of $k$, satisfies the conditions

$$\tag{* } \Omega (tx) = \ \Omega (x) \ \ \textrm{ for } \ t > 0,\ \Omega \in L _ {1} (S),$$

$$\int\limits _ { S } \Omega (x) dS = 0.$$

The Calderón–Zygmund operator is usually written in the form

$$K \phi (x) = \ \textrm{ p.v. } \int\limits _ {\mathbf R ^ {n} } \phi (y) \frac{\Omega (x - y) }{| x - y | ^ {n} } \ dy;$$

here $\textrm{ p }.v.$ denotes the principal value of the integral. In the one-dimensional case the Calderón–Zygmund operator becomes the Hilbert operator $H$:

$$H \phi (x) = \ \textrm{ p.v. } \int\limits _ {- \infty } ^ \infty \frac{\phi (t) }{x - t } dt.$$

The Calderón–Zygmund operator can be extended by continuity to the space $L _ {p} ( \mathbf R ^ {n} )$ of functions $f$ in $\mathbf R ^ {n}$ that are summable of degree $p$ $(1 < p < \infty )$. This extension maps $L _ {p} ( \mathbf R ^ {n} )$ continuously into itself. If $\Omega$ satisfies the condition (*) and also Dini's condition:

$$\int\limits _ { 0 } ^ { 1 } { \frac{\omega (t) dt }{t} } < \infty ,\ \ \omega (t) = \ \sup _ {\begin{array}{c} | x - x ^ \prime | \leq t \\ | x | = | x ^ \prime | = 1 \end{array} } \ | \Omega (x) - \Omega (x ^ \prime ) | ;$$

and if

$$K _ \epsilon f (x) = \ \int\limits _ {| y | > \epsilon } \frac{\Omega (y) }{| y | ^ {n} } f (x - y) dy$$

for $1 < p < \infty$ and $f \in L _ {p} ( \mathbf R ^ {n} )$, then

a) there exists a constant $A _ {p}$( independent of $f$ or $\epsilon$) such that

$$\| K _ \epsilon f \| _ {L _ {p} } \leq \ A _ {p} \| f \| _ {L _ {p} } ;$$

b) the limit $\lim\limits _ {\epsilon \rightarrow 0 } K _ \epsilon f = Kf$ exists in the sense of convergence in $L _ {p}$ and

$$\| Kf \| _ {L _ {p} } \leq \ A _ {p} \| f \| _ {L _ {p} } .$$

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 $L _ {p} ( \mathbf R ^ {n} )$( 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 $\Omega \in L _ {2} (S ^ {n - 1 } )$, [a3], Chapt. VI, Sections 2, 3.

#### 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 caractérisation des opérateurs intégraux singuliers bornés sur $L^2(\RR^n)$" 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://encyclopediaofmath.org/index.php?title=Calder%C3%B3n%E2%80%93Zygmund_operator&oldid=38685