Cauchy integral

A Cauchy integral is a definite integral of a continuous function of one real variable. Let $ f (x) $ be a continuous function on an interval $ [a, b] $ and let $ a = x _ {0} < \dots < x _ {i - 1 } < x _ {i} < \dots < x _ {n} = b $, $ \Delta x _ {i} = x _ {i} - x _ {i - 1 } $, $ i = 1, \dots, n $. The limit

$$ \lim\limits _ {\max \Delta x _ {i} \rightarrow 0 } \ \sum _ {i = 1 } ^ { n } f (x _ {i - 1 } ) \Delta x _ {i} $$

is called the definite integral in Cauchy's sense of $ f (x) $ over $ [a, b] $ and is denoted by

$$ \int\limits _ { a } ^ { b } f (x) dx. $$

The Cauchy integral is a particular case of the Riemann integral. The definition is due to A.L. Cauchy [1].

References[edit]

A Cauchy integral is an integral with the Cauchy kernel,

$$ \frac{1}{2 \pi i ( \zeta - z) } , $$

expressing the values of a regular analytic function $ f (z) $ in the interior of a contour $ L $ in terms of its values on $ L $. More precisely: Let $ f (z) $ be a regular analytic function of the complex variable $ z $ in a domain $ D $ and let $ L $ be a closed piecewise-smooth Jordan curve lying in $ D $ together with its interior $ G $; it is assumed that $ L $ is described in the counter-clockwise sense. Then one has the following formula, which is of fundamental importance in the theory of analytic functions of one complex variable and which is known as the Cauchy integral formula:

$$ \tag{1 } f (z) = \ \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } . $$

The integral on the right of (1) is also called a Cauchy integral.

Apparently, the Cauchy integral first appeared, in certain special cases, in the work of A.L. Cauchy [1].

Cauchy integrals are thus characterized by two conditions: 1) they are evaluated along a closed, smooth (or, at least, piecewise-smooth) curve $ L $; and 2) their integrands have the form

$$ \frac{f ( \zeta ) }{2 \pi i ( \zeta - z) } , $$

where $ \zeta \in L $ and $ f (z) $ is a regular analytic function on $ L $ and in the interior of $ L $. If $ z \in C \overline{G} $ (the complement to $ \overline{G} $) in the Cauchy integral, i.e. if $ z $ lies outside $ L $, then, provided that the conditions 1) and 2) remain valid,

$$ \tag{2 } \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } = 0,\ \ z \in C \overline{G}\; . $$

In particular, if $ L $ is the circle of radius $ \rho $ centred at a point $ z $, i.e.

$$ L = \ \{ {\zeta = z + \rho e ^ {i \theta } } : { 0 \leq \theta < 2 \pi } \} , $$

then (1) implies that

$$ f (z) = \ \frac{1}{2 \pi i } \int\limits _ { 0 } ^ { {2 } \pi } f (z + \rho e ^ {i \theta } ) \ d \theta , $$

i.e. the value of $ f (z) $ at any point $ z \in D $ is equal to the arithmetic average of its values on any sufficiently small circle $ L \subset D $ centred at $ z $. Formula (1) enables one to prove all other elementary properties of analytic functions.

On the other hand, if $ f (z) $ is a regular analytic function in the infinite domain $ C \overline{G} $ (the exterior of the closed curve $ L $) and on $ L $, and if one defines

$$ f ( \infty ) = \ \lim\limits _ {z \rightarrow \infty } f (z), $$

then the following formula, known as the Cauchy integral formula for an infinite domain, is valid:

$$ \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } = \ \left \{ \begin{array}{ll} f ( \infty ) - f (z), &z \in C \overline{G}\; , \\ f ( \infty ), &z \in G. \\ \end{array} \right .$$

Now let $ \Gamma $ be some (not necessarily closed) piecewise-smooth curve in the finite plane, $ z \neq \infty $, let $ \phi ( \zeta ) $ be a continuous complex function on $ \Gamma $ and let $ z $ be a point not on $ \Gamma $. The term integral of Cauchy type is applied to the following generalization of the Cauchy integral:

$$ \tag{3 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - z } ,\ \ z \notin \Gamma . $$

The function $ \phi ( \zeta ) $ is called the density of the integral of Cauchy type. Elementary properties of integrals of Cauchy type are:

1) $ F (z) $ is a regular analytic function of $ z $ in any domain not containing points of $ \Gamma $.

2) The derivatives $ F ^ { (n) } (z) $ are given by the formulas

$$ F ^ { (n) } (z) = \ \frac{n! }{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{( \zeta - z) ^ {n + 1 } } ,\ \ z \notin \Gamma ; \ \ n = 0, 1 , . . . . $$

3) $ F (z) $ is regular at infinity, with $ F ( \infty ) = 0 $, $ F (z) = O (1/z) $ as $ z \rightarrow \infty $.

From the point of view of the general theory of analytic functions and its applications to mechanics and physics, it is of fundamental importance to consider the existence of boundary values of an integral of Cauchy type as one approaches $ \Gamma $, and to find analytic expressions for these values. The Cauchy integral (1) is equal to $ f (z) $ everywhere in the interior of $ L $ and vanishes identically outside $ L $. Therefore, when an integral of Cauchy type (3) reduces to a Cauchy integral, i.e. when the conditions 1) and 2) are satisfied, then, as $ L $ is approached from the left (i.e. from its interior), the function $ F (z) $ has boundary values $ F ^ {+} ( \zeta _ {0} ) = f ( \zeta _ {0} ) $, and if these values are assumed on $ L $ it is continuous from the left on $ L $ at each point $ \zeta _ {0} \in L $; as $ L $ is approached from the right (i.e. from its exterior), then $ F (z) $ has boundary values zero, i.e. $ F ^ {-} ( \zeta _ {0} ) = 0 $, and if these values are assumed on $ L $ it is continuous from the right on $ L $ at each point $ \zeta _ {0} \in L $. Thus, for a Cauchy integral

$$ F ^ {+} ( \zeta _ {0} ) - F ^ {-} ( \zeta _ {0} ) = \ f ( \zeta _ {0} ). $$

For an integral of Cauchy type of general form the matter is somewhat more complicated. Suppose that the equation of the curve $ \Gamma $ is $ \zeta = \zeta (s) $, where $ s $ denotes the arc length reckoned from some fixed point, let $ \zeta _ {0} = \zeta (s _ {0} ) $ be an arbitrary fixed point on $ \Gamma $ and let $ \Gamma _ \epsilon $ be the part of $ \Gamma $ that remains after the smaller of the arcs with end points $ \zeta (s _ {0} - \epsilon ) $ and $ \zeta (s _ {0} + \epsilon ) $ is deleted from $ \Gamma $. If the limit

$$ \tag{4 } \lim\limits _ {\epsilon \rightarrow 0 } \frac{1}{2 \pi i } \int\limits _ {\Gamma _ \epsilon } \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } ,\ \ \zeta _ {0} \in \Gamma , $$

exists and is finite, it is called a singular integral. It can be proved, for example, that a singular integral (4) exists if the curve $ \Gamma $ is smooth in a neighbourhood of a point $ \zeta _ {0} $ distinct from the end points of $ \Gamma $ and if the density $ \phi ( \zeta ) $ satisfies a Hölder condition

$$ | \phi ( \zeta ^ \prime ) - \phi ( \zeta ^ {\prime\prime} ) | \leq \ C | \zeta ^ \prime - \zeta ^ {\prime\prime} | ^ \mu ,\ \ \mu > 0. $$

Under these conditions there also exist boundary values, and these are given by the Sokhotskii formulas:

$$ \tag{5 } F ^ { \pm } ( \zeta _ {0} ) = \ \pm { \frac{1}{2} } \phi ( \zeta _ {0} ) + \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } ,\ \ \zeta _ {0} \in \Gamma , $$

and the functions $ F ^ {+} (z) $ and $ F ^ {-} (z) $ are continuous in a neighbourhood of $ \zeta _ {0} \in \Gamma $ from the left and right, respectively, of $ \Gamma $. In the case of a Cauchy integral, the singular integral is equal to

$$ \frac{f ( \zeta _ {0} ) }{2} , $$

$$ F ^ {+} ( \zeta _ {0} ) - F ^ {-} ( \zeta _ {0} ) = \ f ( \zeta _ {0} ) ,\ F ^ {+} ( \zeta _ {0} ) + F ^ {-} ( \zeta _ {0} ) = f ( \zeta _ {0} ). $$

An equivalent form of (5) is

$$ \tag{6 } F ^ {+} ( \zeta _ {0} ) - F ^ {-} ( \zeta _ {0} ) = \ \phi ( \zeta _ {0} ), $$

$$ \tag{7 } F ^ {+} ( \zeta _ {0} ) + F ^ {-} ( \zeta _ {0} ) = \frac{1}{\pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } ,\ \zeta _ {0} \in \Gamma . $$

The Sokhotskii formulas (5)–(7) are of fundamental importance in the solution of boundary value problems of analytic function theory, of singular integral equations connected with integrals of Cauchy type (cf. Singular integral equation), and also in the solution of various problems in hydrodynamics, elasticity theory, etc.

Let $ \Gamma $ be an arbitrary rectifiable curve of length $ l $; for simplicity it is assumed that $ \Gamma $ is closed. Let $ \psi = \psi (s) $ be the angle between the direction of the $ x $-axis and the tangent to $ \Gamma $ at the point $ \zeta = \zeta (s) \in \Gamma $, regarded as a function of the arc length $ s $, and let $ \Phi (s) $ be a complex function of $ s $ of bounded variation on $ [0, l] $. The expression

$$ \tag{8 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - z } ,\ \ \zeta = \zeta (s),\ \ z \notin \Gamma , $$

is called an integral of Cauchy–Stieltjes type. In other words, an integral of Cauchy–Stieltjes type is an integral of Cauchy type with respect to an arbitrary finite complex Borel measure with support on $ \Gamma $. If $ \Phi (s) $ is absolutely continuous, then the integral of Cauchy–Stieltjes type becomes an integral of Cauchy–Lebesgue type, often called simply an integral of Cauchy type:

$$ \tag{9 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - z } ,\ \ z \notin \Gamma , $$

where $ \phi ( \zeta ) = \phi [ \zeta (s)] = \Phi ^ \prime (s) $.

Let $ \zeta _ {0} $ be a point of $ \Gamma $ at which there exists a well-defined tangent, inclined to the $ x $-axis at an angle $ \psi _ {0} $; such points exist almost-everywhere on a rectifiable curve. Let $ z $ be the point on the straight line passing through $ \zeta _ {0} $ and inclined to the normal at an angle $ \alpha _ {0} $, at a distance $ | z - \zeta _ {0} | = \epsilon $, i.e. $ z = \zeta _ {0} \pm \epsilon i e ^ {i ( \psi _ {0} + \alpha _ {0} ) } $. The difference between the integral of Cauchy–Stieltjes type (8) and the integral over $ \Gamma _ \epsilon $,

$$ W ( \zeta _ {0} ; \epsilon , \alpha _ {0} ) = \ \frac{1}{2 \pi i } \left [ \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - z } - \int\limits _ {\Gamma _ \epsilon } \frac{e ^ {i \psi } d \Phi (s) }{\zeta - \zeta _ {0} } \right ] , $$

is defined at all points $ \zeta _ {0} \in \Gamma $ where the tangent is defined, i.e. almost-everywhere on $ \Gamma $. An important proposition in the theory of integrals of Cauchy–Stieltjes type is Privalov's fundamental lemma: The limit

$$ \lim\limits _ {\epsilon \rightarrow 0 } \ W ( \zeta _ {0} ; \epsilon , \alpha _ {0} ) = \ \pm { \frac{1}{2} } \Phi ^ \prime ( \zeta _ {0} ) $$

exists for all points $ \zeta _ {0} \in \Gamma $, with the possible exception of a point set of measure zero on $ \Gamma $, independent of $ \alpha _ {0} $; the convergence is uniform in $ \alpha _ {0} $ in any angle $ | \alpha _ {0} | < \pi /2 - \delta $, $ \delta > 0 $. If the singular integral exists almost-everywhere on $ \Gamma $, then the integral of Cauchy–Stieltjes type has angular boundary values $ F ^ { \pm } ( \zeta _ {0} ) $ almost-everywhere on $ \Gamma $ and these satisfy the Sokhotskii formulas:

$$ \tag{10 } F ^ \pm ( \zeta _ {0} ) = \ \pm { \frac{1}{2} } \Phi ^ \prime (s _ {0} ) + \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - \zeta _ {0} } ,\ \ \zeta _ {0} \in \Gamma . $$

The converse is also true: If an integral of Cauchy–Stieltjes type has angular boundary values from both inside and outside $ \Gamma $, almost-everywhere on $ \Gamma $, then the singular integral exists and formulas (10) are valid almost-everywhere on $ \Gamma $. As yet (1987) there is no complete solution to the problem of finding reasonably simple necessary and sufficient conditions for the existence of boundary values for integrals of Cauchy–Stieltjes type or even for integrals of Cauchy–Lebesgue type.

In contrast to the previously considered case of an integral of Cauchy type over a smooth curve $ \Gamma $, an integral of Cauchy–Stieltjes type, even when it has angular boundary values, is no longer necessarily a continuous function in a neighbourhood of $ \zeta _ {0} \in \Gamma $ from the left or right of $ \Gamma $. It is known, for example, that an integral of Cauchy–Lebesgue type (9) is continuous in the closed domain $ \overline{D}\; $ bounded by the rectifiable contour $ \Gamma $, provided one additionally assumes that the density $ \phi ( \zeta ) $ satisfies a Lipschitz condition on $ \Gamma $:

$$ | \phi ( \zeta ^ \prime ) - \phi ( \zeta ^ {\prime\prime} ) | \leq C | \zeta ^ \prime - \zeta ^ {\prime\prime} |,\ \ \zeta ^ \prime ,\ \zeta ^ {\prime\prime} \in \Gamma . $$

One says that an integral of Cauchy–Lebesgue type (9) becomes a Cauchy integral

$$ \tag{11 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - z } , $$

in the sense of Lebesgue, if the angular boundary values $ F ^ {+} ( \zeta _ {0} ) $ from the inside of $ \Gamma $ coincide with $ \phi ( \zeta _ {0} ) $ almost-everywhere on $ \Gamma $, i.e. $ F ^ {-} ( \zeta _ {0} ) = 0 $ almost-everywhere on $ \Gamma $. In this context the Golubev–Privalov theorem holds: A summable function $ \phi ( \zeta ) $ on $ \Gamma $ represents the angular boundary values of some Cauchy integral from the inside of $ \Gamma $ if and only if all its moments vanish:

$$ \tag{12 } \int\limits _ \Gamma \zeta ^ {n} \phi ( \zeta ) \ d \zeta = 0,\ \ n = 0, 1 , . . . . $$

If the analogous conditions

$$ \tag{13 } \int\limits _ \Gamma \zeta ^ {n} e ^ {i \psi } d \Phi (s) = 0,\ \ n = 0, 1, \dots $$

are satisfied, then the integral of Cauchy–Stieltjes type (8) becomes a Cauchy–Stieltjes integral:

$$ \tag{14 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - z } , $$

i.e. the angular boundary values $ F ^ {+} ( \zeta _ {0} ) $ from the inside of $ \Gamma $ coincide with the derivative $ \Phi ^ \prime (s _ {0} ) $ almost-everywhere on $ \Gamma $, or, stated differently, the angular boundary values $ F ^ {-} ( \zeta _ {0} ) $ from the outside of $ \Gamma $ vanish almost-everywhere on $ \Gamma $. Conditions (13) immediately imply that the function $ \Phi (s) $ is absolutely continuous on $ [0, l] $ and, consequently, in this case the Cauchy–Stieltjes integral (14) is in fact a Cauchy–Lebesgue integral with density $ \phi ( \zeta ) = \phi [ \zeta (s)] = \Phi ^ \prime (s) $. Thus, the class of functions representable by a Cauchy–Stieltjes integral is identical with the class of functions representable by a Cauchy–Lebesgue integral.

An important problem is the intrinsic characterization of classes of functions which are regular in a domain $ D $ bounded by a closed rectifiable curve $ \Gamma $, and representable by a Cauchy integral (11), an integral of Cauchy–Lebesgue type (9), or an integral of Cauchy–Stieltjes type (8); concerning the most important classes $ A (D) $, $ B (D) = H _ \infty (D) $, $ H _ {p} (D) $ and $ N ^ {*} (D) $ see Boundary properties of analytic functions.

In the simplest case, when $ D = \{ {z } : {| z | < 1 } \} $ is the unit disc and $ \Gamma = \{ {z } : {| z | = 1 } \} $ is the unit circle, an integral of Cauchy–Stieltjes type, which in this case has the form

$$ \tag{15 } F (z) = \ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \frac{\zeta d \Phi ( \theta ) }{\zeta - z } ,\ \ | z | < 1,\ \ \zeta = e ^ {i \theta } , $$

always represents a function of class $ H _ {p} $, $ 0 < p < 1 $. The converse is false: The set of functions of classes $ H _ {p} $, $ 0 < p < 1 $, is more extensive than the set of functions representable in the form (15). On the other hand, the set of functions representable in $ D $ by a Cauchy–Stieltjes or a Cauchy integral is identical with the class $ H _ {1} $.

In the case of an arbitrary simply-connected domain $ D $ bounded by a rectifiable curve $ \Gamma $, the class of functions representable in $ D $ by a Cauchy–Stieltjes or a Cauchy integral is identical with the Smirnov class $ E _ {1} $ (see Boundary properties of analytic functions). The characteristics of the classes of functions representable by an integral of Cauchy–Stieltjes type or an integral of Cauchy–Lebesgue type are considerably more complicated.

Let $ f (z) $ be an arbitrary (non-analytic) function of class $ C ^ {1} $ in a finite closed domain $ \overline{D}\; $ bounded by a piecewise-smooth Jordan curve $ L $. The term Cauchy integral formula is sometimes applied also to the following generalization of the classical formula (1):

$$ \tag{16 } \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } - { \frac{1} \pi } {\int\limits \int\limits } _ { D } \frac{\partial f }{\partial \overline \zeta \; } \frac{d \xi d \eta }{\zeta - z } = $$

$$ = \ \left \{ \begin{array}{ll} f (z), & z \in D, \\ 0, & z \in C \overline{D}\; , \\ \end{array} \right .$$

where

$$ \frac{\partial f }{\partial \overline \zeta \; } = \ { \frac{1}{2} } \left ( \frac{\partial f }{\partial \xi } + i \frac{\partial f }{\partial \eta } \right ) ,\ \ \zeta = \xi + i \eta . $$

This formula first appeared, apparently, in the work of D. Pompeiu (1912). It is also known as the Pompeiu formula, the Borel–Pompeiu formula, or the Cauchy–Green formula, and is widely applied in the theory of generalized analytic functions, singular integral equations and various applied problems.

Let $ f (z) $ be a regular analytic function of several complex variables $ z = (z _ {1}, \dots, z _ {n} ) $ in a closed polydisc $ \overline{D} $, $ D = \{ {z \in \mathbf C ^ {n} } : {| z _ \nu - a _ \nu | < r _ \nu } \} $. Then, at each point of $ D $, $ f (z) $ is representable by a multiple Cauchy integral:

$$ \tag{17 } f (z) = \ \frac{1}{(2 \pi i) ^ {n} } \int\limits _ { T } \frac{f ( \zeta ) d \zeta }{\zeta - z } , $$

where $ T = \{ {\zeta \in \mathbf C ^ {n} } : {| \zeta _ \nu - a _ \nu | = r _ \nu , \nu = 1, \dots, n } \} $ is the distinguished boundary of the polydisc, $ \zeta = ( \zeta _ {1}, \dots, \zeta _ {n} ) $, $ d \zeta = d \zeta _ {1} \dots d \zeta _ {n} $, $ \zeta - z = ( \zeta _ {1} - z _ {1} ) \dots ( \zeta _ {n} - z _ {n} ) $. Formula (17) yields a simple analogue of the Cauchy integral for a circle $ L = \{ {z \in \mathbf C } : {| z - a | = r } \} $, but when $ n > 1 $ the integration in (17) extends not over the entire boundary of the polydisc but only over its distinguished boundary. In general, let $ D = D _ {1} \times \dots \times D _ {n} $ be a polycircular domain in $ \mathbf C ^ {n} $— a product of simply-connected plane domains $ D _ \nu $ with smooth boundaries $ \partial D _ \nu = \{ {z _ \nu = z _ \nu ( t _ \nu ) } : {0 \leq t _ \nu \leq 1 } \} $; let $ T = \partial D _ {1} \times \dots \times \partial D _ {n} $ be the distinguished boundary of $ D $, which is a smooth $ n $-dimensional manifold. Formula (17) also generalizes to this case.

More profound generalizations of the Cauchy integral formula are extremely important in the theory of analytic functions of several complex variables; such generalizations are the Leray formula (which J. Leray himself called the Cauchy–Fantappié formula) and the Bochner–Martinelli representation formula. In this connection, when $ n > 1 $ the theory is concerned mainly with boundary properties of integral representations other than (17).

References[edit]

Comments[edit]

In the non-Soviet literature Plemelj formulas is the usual name for what is here called Sokhotskii formulas.

Mapping properties of the singular integral operator associated to integrals of Cauchy type form an important subject. Let $ \Gamma $ be the graph of a Lipschitz function $ \phi (x) $. The principal result, due to A.P. Calderón and in full generality to G. David, is that the singular integral operator

$$ f \rightarrow \ \int\limits _ \Gamma \frac{f ( \zeta ) }{z - \zeta } \ d \zeta , $$

at first defined as a principal value integral for compactly supported smooth functions $ f $ on $ \Gamma $, extends to a bounded linear operator sending $ L _ {2} ( \Gamma ) $ to itself, and (hence) also sending $ L _ {p} ( \Gamma ) $ to itself $ (1 < p < \infty ) $ and $ L _ \infty ( \Gamma ) $ to $ \mathop{\rm BMO} $, the functions of bounded mean oscillation.

Formally one can write:

$$ \int\limits _ \Gamma \frac{f ( \zeta ) }{z - \zeta } \ d \zeta = \ \int\limits _ {\mathbf R } \frac{f ( \xi + i \phi ( \xi )) }{x - \xi + i ( \phi (x) - \phi ( \xi )) } (1 + \phi ^ \prime ( \xi )) \ d \xi = $$

$$ = \ \sum _ { 0 } ^ \infty \int\limits _ {\mathbf R } \frac{f ( \xi + i \phi ( \xi )) (1 + i \phi ( \xi )) }{x - \xi } \left ( \frac{(-i) ( \phi (x) - \phi ( \xi )) }{x - \xi } \right ) ^ {n} d \xi . $$

The integral operators $ C _ {n} ( \phi ) $ with kernel

$$ { \frac{1}{x - \xi } } \left ( \frac{\phi (x) - \phi ( \xi ) }{x - \xi } \right ) ^ {n} $$

are the so-called commutators of Calderón. These are of independent interest, e.g. in the theory of partial differential equations (cf. Differential equation, partial). The operators $ C _ {n} ( \phi ) $ have the same mapping properties as the Cauchy integral operator, as was shown by R.R. Coifman, A. McIntosh and Y. Meyer. The best norm estimate known at this moment (1987) is that for every $ \delta > 0 $ there exists a $ c _ \delta > 0 $ such that

$$ \| C _ {n} ( \phi ) \| \leq \ c _ \delta (n + 1) ^ {1 + \delta } \ \| \phi ^ \prime \| _ \infty ^ {n} . $$

This estimate was obtained by M. Christ and J.L. Journé.

The Cauchy integral operators as well as Calderón's commutators are examples of so-called Calderón–Zygmund operators. For these results and further references see [10], [a2], [a3], [a4].

For results concerning $ H _ {p} $ functions, $ 0 < p < 1 $, which can be represented by Cauchy integrals, see [a1].

References[edit]