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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|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 [[#References|[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|Bochner–Martinelli representation formula]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  J. Leray, "Le calcul différentiel et intégrale sur une variété analytique complexe"  ''Bull. Soc. Math. France'' , '''87'''  (1959)  pp. 81–180</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR>
</table>

====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 [[#References|[a2]]]).

Thus, the Leray formula has become an important tool for solving the [[Levi problem|Levi problem]] (work of G.M. Khenkin [[#References|[a1]]] and of E. Ramirez de Arellano [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Leiterer,   "Theory of functions on complex manifolds" , Birkhäuser  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Ramirez de Arellano,   "Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis"  ''Math. Ann.'' , '''184'''  (1970)  pp. 172–187</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. VI, Par. 6</TD></TR></table>