Leray formula

Cauchy–Fantappié formula

A formula for the integral representation of holomorphic functions $ f ( z) $ of several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n \geq 1 $, which generalizes the Cauchy integral formula (see Cauchy integral).

Let $ D $ be a finite domain in the complex space $ \mathbf C ^ {n} $ with piecewise-smooth boundary $ \partial D $ and let $ \chi ( \zeta ; z ) : \partial D \rightarrow \mathbf C ^ {n} $ be a smooth vector-valued function of $ \zeta \in \partial D $ with values in $ \mathbf C ^ {n} $ such that the scalar product

$$ \langle \zeta - z , \chi ( \zeta ; z ) \rangle = \sum _ {\nu = 1 } ^ { n } ( \zeta _ \nu - z _ \nu ) \chi _ \nu ( \zeta ; z ) \neq 0 $$

everywhere on $ \partial D $ for all $ z \in D $. Then any function $ f ( z) $ holomorphic in $ D $ and continuous in the closed domain $ \overline{D}\; $ can be represented in the form

$$ \tag{* } f ( z) = \frac{( n- 1 )! }{( 2 \pi i ) ^ {n} } \int\limits _ {\partial D } \frac{f ( \zeta ) \delta ( \chi ( \zeta ; z )) \wedge d \zeta }{< \zeta - z , \chi ( \zeta ; z ) > ^ {n} } ,\ z \in D . $$

Formula (*) generalizes Cauchy's classical integral formula for analytic functions of one complex variable and is called the Leray formula. J. Leray, who obtained this formula (see [1]), called it the Cauchy–Fantappié formula. In this formula the differential forms $ \delta ( \chi ( \zeta ; z )) $ and $ d \zeta $ are constituted according to the laws:

$$ \delta ( \chi ( \zeta ; z )) = \sum _ {\nu = 1 } ^ { n- } 1 ( - 1 ) ^ { \nu - 1 } \chi _ \nu ( \zeta ; z ) d \chi _ {1} ( \zeta ; z ) \wedge \dots $$

$$ \dots \wedge d \chi _ {\nu - 1 } ( \zeta ; z ) \wedge d \chi _ {\nu + 1 } ( \zeta ; z ) \wedge \dots \wedge d \chi _ {n} ( \zeta ; z ) $$

and

$$ d \zeta = d \zeta _ {1} \wedge \dots \wedge d \zeta _ {n} , $$

where $ \wedge $ is the sign of exterior multiplication (see Exterior product). By varying the form of the function $ \chi $ it is possible to obtain various integral representations from formula (*). One should bear in mind that, generally speaking, the Leray integral in (*) is not identically zero when $ z $ is outside $ D $.

See also Bochner–Martinelli representation formula.

References

[1]	J. Leray, "Le calcul différentielle et intégrale sur une variété analytique complexe" Bull. Soc. Math. France , 87 (1959) pp. 81–180
[2]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)

Comments

Often the Leray formula is understood to be a more general representation formula, valid for arbitrary sufficiently smooth (e.g., $ C ^ {1} $) functions on a domain $ D $ in $ \mathbf C ^ {n} $. Let $ \chi ( \zeta , z ) $, $ \delta $ and $ d $ be as defined above, $ \psi ( \zeta , z ) = \langle \zeta - z , \chi ( \zeta , z ) \rangle $. Furthermore, define for $ z \in D $, $ \zeta \in \partial D $ and $ 0 \leq \lambda \leq 1 $:

$$ \eta ^ \chi ( z , \zeta , \lambda ) = \ ( 1 - \lambda ) \frac{\chi ( \zeta , z ) }{\psi ( \zeta , z ) } + \lambda \frac{( \overline \zeta \; - \overline{z}\; ) }{\| \zeta - z \| ^ {2} } . $$

Let $ L _ {\partial D } ^ \chi f ( z) $ denote the right-hand side of (*). It is well defined for measurable functions $ f $ on $ \partial D $. Define for a continuous $ 1 $- form $ u $ on $ \partial D $,

$$ R _ {\partial D } ^ \chi u ( z) = \ \frac{( n - 1 ) ! }{( 2 \pi i ) ^ {n} } \int\limits _ {\begin{array}{c} \zeta \in \partial D \\ 0 \leq \lambda \leq 1 \end{array} } u \wedge \delta _ {\zeta , \lambda } ( \eta ) \wedge d \zeta , $$

$ \delta _ {\zeta , \lambda } $ meaning that the exterior derivative in the definition of $ \delta $ has to be with respect to $ \zeta $ as well as $ \lambda $. Next, for $ 1 $- forms $ u $ defined on $ D $ there holds

$$ B _ {D} u ( z) = \ \frac{( n - 1 ) ! }{( 2 \pi i ) ^ {n} } \int\limits _ {\zeta \in \partial D } u \wedge \delta _ \zeta \left ( \frac{\overline \zeta \; - \overline{z}\; }{\| \zeta - z \| ^ {2} } \right ) \wedge d \zeta , $$

the Bochner–Martinelli operator.

Now let $ f $ be a continuous function on $ \overline{D}\; $ such that $ \overline \partial \; f $ is continuous there too. Then Leray's formula reads

$$ \tag{a1 } f ( z) = L _ {\partial D } ^ \chi f ( z) - R _ {\partial D } ^ \chi \overline \partial \; f ( z) - B _ {D} \overline \partial \; f ( z) , $$

where $ z \in D $.

If $ f $ is holomorphic on $ D $, then (a1) reduces to (*). Of particular importance are instances where $ \chi $, and hence also $ \psi $, is holomorphic as a function of $ z $ for $ \zeta $ fixed — this can only occur if $ D $ is pseudo-convex; $ \psi $ is then a holomorphic support function (i.e. for all $ p \in \partial D $ there is a neighbourhood $ U _ {p} $ of $ p $ such that $ \psi $ is holomorphic in this neighbourhood and $ \{ {z \in U _ {p} } : {\psi ( z ) = 0 } \} \cap \overline{D}\; = \{ p \} $), the existence of which is closely related to the existence of continuously varying holomorphic peaking functions. (A continuously varying holomorphic peaking function for $ D $ is a function $ P : \overline{D}\; \times \partial D \rightarrow \mathbf C $ such that for each fixed $ p \in \partial D $: 1) $ P ( \cdot , p ) $ is holomorphic on $ D $ and continuous on $ \overline{D}\; $; and 2) $ P ( p , p ) = 1 $ and $ | P ( z , p ) | < 1 $ for all $ z \in \overline{D}\; \setminus \{ p \} $. If $ \partial D \in C ^ {k+} 3 $, $ P ( z , \cdot ) $ is required to be $ C ^ {k} $ for each fixed $ z \in D $.) Then $ L _ {\partial D } ^ \chi f $ is holomorphic for every continuous $ f $ on $ \partial D $ and the operator

$$ u \mapsto f = - ( R _ {\partial D } ^ \chi u + B _ {D} u ) $$

solves the inhomogeneous Cauchy–Riemann equations

$$ \tag{a2 } \left . \begin{array}{c} \overline \partial \; f = u \\ \textrm{ with integrability condition } \overline \partial \; u = 0 \\ \end{array} \right \} $$

for continuous $ ( 0 , 1 ) $- forms $ u $ on $ \overline{D}\; $. Formula (a1) can be generalized to give a representation formula for $ ( p , q ) $- forms as well (see [a2]).

Thus, the Leray formula has become an important tool for solving the Levi problem (work of G.M. Khenkin [a1] and of E. Ramirez de Arellano [a3]) and for obtaining estimates for solutions of (a2). In particular, the following sharp Hölder estimates hold on strictly pseudo-convex domains: There is a solution $ f $ with $ \| f \| _ {1/2} \leq C \| u \| _ \infty $, where $ C $ depends on the domain only, $ \| \cdot \| _ {1/2} $ denotes the Hölder $ 1/2 $- norm and $ \| \cdot \| _ \infty $ denotes the sup-norm. Many analysts made contributions in this direction, notably Khenkin and A.V. Romanov; H. Grauert and I. Lieb; and N. Kerzman and R.M. Range.

References

[a1]	G.M. [G.M. Khenkin] Henkin, "Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications" Math. USSR Sb. , 78 (1969) pp. 611–632 Mat. Sb. , 7 (1969) pp. 597–616
[a2]	J.L. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984)
[a3]	E. Ramirez de Arellano, "Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis" Math. Ann. , 184 (1970) pp. 172–187
[a4]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. VI, Par. 6