Leray formula

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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 $D$.

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)

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
How to Cite This Entry:
Leray formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leray_formula&oldid=47615
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article