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An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200701.png" /> defined on a space of sufficiently smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200702.png" /> with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200703.png" /> by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200704.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200705.png" /> is a homogeneous function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200706.png" /> with zero mean value over the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200707.png" />. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200708.png" /> has the form
+
An operator  $  K $
 +
defined on a space of sufficiently smooth functions  $  \phi $
 +
with compact support in  $  \mathbf R  ^ {n} $
 +
by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c0200709.png" /></td> </tr></table>
+
$$
 +
K \phi (x)  = \
 +
\lim\limits _ {\epsilon \rightarrow 0 } \
 +
\int\limits _ {| x - y | > \epsilon }
 +
k (x - y) \phi (y)  dy,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007010.png" />, the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007011.png" />, satisfies the conditions
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
k (x) = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007013.png" /></td> </tr></table>
+
\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
 
The Calderón–Zygmund operator is usually written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007014.png" /></td> </tr></table>
+
$$
 +
K \phi (x)  = \
 +
\textrm{ p.v. }
 +
\int\limits _ {\mathbf R  ^ {n} }
 +
\phi (y)
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007015.png" /> denotes the principal value of the integral. In the one-dimensional case the Calderón–Zygmund operator becomes the Hilbert operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007016.png" />:
+
\frac{\Omega (x - y) }{| x - y |  ^ {n} }
 +
\
 +
dy;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007017.png" /></td> </tr></table>
+
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 $:
  
The Calderón–Zygmund operator can be extended by continuity to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007018.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007020.png" /> that are summable of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007021.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007022.png" />. This extension maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007023.png" /> continuously into itself. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007024.png" /> satisfies the condition (*) and also Dini's condition:
+
$$
 +
H \phi (x)  = \
 +
\textrm{ p.v. }
 +
\int\limits _ {- \infty } ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007025.png" /></td> </tr></table>
+
\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
 
and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007026.png" /></td> </tr></table>
+
$$
 +
K _  \epsilon  f (x)  = \
 +
\int\limits _ {| y | > \epsilon }
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007028.png" />, then
+
\frac{\Omega (y) }{| y |  ^ {n} }
  
a) there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007029.png" /> (independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007030.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007031.png" />) such that
+
f (x - y) dy
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007032.png" /></td> </tr></table>
+
for  $  1 < p < \infty $
 +
and  $  f \in L _ {p} ( \mathbf R  ^ {n} ) $,
 +
then
  
b) the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007033.png" /> exists in the sense of convergence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007034.png" /> and
+
a) there exists a constant  $  A _ {p} $(
 +
independent of $  f $
 +
or  $  \epsilon $)
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007035.png" /></td> </tr></table>
+
$$
 +
\| K _  \epsilon  f \| _ {L _ {p}  }  \leq  \
 +
A _ {p} \| f \| _ {L _ {p}  } ;
 +
$$
 +
 
 +
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 [[#References|[1]]].
 
The Calderón–Zygmund operator was analyzed by A.P. Calderón and A. Zygmund [[#References|[1]]].
Line 43: Line 139:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Calderón,  A. Zygmund,  "On the existence of certain singular integrals"  ''Acta Math.'' , '''88'''  (1952)  pp. 85–139</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Mikhlin,  "Multidimensional singular integrals and integral equations" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Calderón,  A. Zygmund,  "On the existence of certain singular integrals"  ''Acta Math.'' , '''88'''  (1952)  pp. 85–139</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Mikhlin,  "Multidimensional singular integrals and integral equations" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Proofs of the estimates a) and b) above can be found in [[#References|[3]]], Chapt. II, Section 4.
 
Proofs of the estimates a) and b) above can be found in [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007036.png" /> (estimate a)) was proved by S.G. Mikhlin in 1938 (published in [[#References|[a1]]]).
+
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 [[#References|[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 [[#References|[a2]]].
 
Nowadays the term Calderón–Zygmund operator is often used for general operators in the class defined by G. David and J.L. Journé in [[#References|[a2]]].
  
Assertion b) has been generalized to singular integrals with odd kernels, and to singular integrals with even kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020070/c02007037.png" />, [[#References|[a3]]], Chapt. VI, Sections 2, 3.
+
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 } ) $,  
 +
[[#References|[a3]]], Chapt. VI, Sections 2, 3.
  
 
See also [[Singular integral|Singular integral]]; [[Hilbert singular integral|Hilbert singular integral]]; [[Hilbert transform|Hilbert transform]].
 
See also [[Singular integral|Singular integral]]; [[Hilbert singular integral|Hilbert singular integral]]; [[Hilbert transform|Hilbert transform]].

Revision as of 06:29, 30 May 2020


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.

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://encyclopediaofmath.org/index.php?title=Calder%C3%B3n-Zygmund_operator&oldid=46187
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article