Difference between revisions of "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 | + | 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 | 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 | 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 [[#References|[1]]]. | The Calderón–Zygmund operator was analyzed by A.P. Calderón and A. Zygmund [[#References|[1]]]. | ||
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====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 | + | 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 | + | 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 ![]() |
[a3] | E.M. Stein, G. Weiss, "Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1975) |
Calderón-Zygmund operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calder%C3%B3n-Zygmund_operator&oldid=23210