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''Cauchy–Fantappié formula''
 
''Cauchy–Fantappié formula''
  
A formula for the integral representation of holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581801.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581803.png" />, which generalizes the Cauchy integral formula (see [[Cauchy integral|Cauchy integral]]).
+
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
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581804.png" /> be a finite domain in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581805.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581806.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581807.png" /> be a smooth vector-valued function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581808.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l0581809.png" /> such that the scalar product
+
$$
 +
\langle  \zeta - z , \chi ( \zeta ;  z ) \rangle  = \sum _ {\nu = 1 } ^ { n }
 +
( \zeta _  \nu  - z _  \nu  ) \chi _  \nu  ( \zeta ;  z )  \neq  0
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818010.png" /></td> </tr></table>
+
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
  
everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818011.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818012.png" />. Then any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818013.png" /> holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818014.png" /> and continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818015.png" /> can be represented in the form
+
$$ \tag{* }
 +
f ( z)  =
 +
\frac{( n- 1 )! }{( 2 \pi i )  ^ {n} }
 +
\int\limits _ {\partial  D }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818018.png" /> are constituted according to the laws:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818019.png" /></td> </tr></table>
+
$$
 +
\delta ( \chi ( \zeta ; z ))  = \sum _ {\nu = 1 } ^ { n- }  1 ( - 1 ) ^ {
 +
\nu - 1 } \chi _  \nu  ( \zeta ; z )  d \chi _ {1} ( \zeta ; z ) \wedge \dots
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818020.png" /></td> </tr></table>
+
$$
 +
\dots \wedge d \chi _ {\nu - 1 }  ( \zeta ; z ) \wedge d \chi _ {\nu
 +
+ 1 }  ( \zeta ; z ) \wedge \dots \wedge d \chi _ {n} ( \zeta ; z )
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818021.png" /></td> </tr></table>
+
$$
 +
d \zeta  = d \zeta _ {1} \wedge \dots \wedge d \zeta _ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818022.png" /> is the sign of exterior multiplication (see [[Exterior product|Exterior product]]). By varying the form of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818023.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818024.png" /> is outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818025.png" />.
+
where $  \wedge $
 +
is the sign of exterior multiplication (see [[Exterior product|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]].
 
See also [[Bochner–Martinelli representation formula|Bochner–Martinelli representation formula]].
Line 28: Line 78:
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Leray,  "Le calcul différentielle 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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Leray,  "Le calcul différentielle 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)  = \
  
====Comments====
+
\frac{( n - 1 ) ! }{( 2 \pi i )  ^ {n} }
Often the Leray formula is understood to be a more general representation formula, valid for arbitrary sufficiently smooth (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818026.png" />) functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818031.png" /> be as defined above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818032.png" />. Furthermore, define for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818035.png" />:
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818036.png" /></td> </tr></table>
+
\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 ,
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818037.png" /> denote the right-hand side of (*). It is well defined for measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818039.png" />. Define for a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818040.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818041.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818042.png" />,
+
$  \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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818043.png" /></td> </tr></table>
+
$$
 +
B _ {D} u ( z)  = \
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818044.png" /> meaning that the exterior derivative in the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818045.png" /> has to be with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818046.png" /> as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818047.png" />. Next, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818048.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818049.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818050.png" /> there holds
+
\frac{( n - 1 ) ! }{( 2 \pi i )  ^ {n} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818051.png" /></td> </tr></table>
+
\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.
 
the Bochner–Martinelli operator.
  
Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818052.png" /> be a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818053.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818054.png" /> is continuous there too. Then Leray's formula reads
+
Now let $  f $
 +
be a continuous function on $  \overline{D}\; $
 +
such that $  \overline \partial \; f $
 +
is continuous there too. Then Leray's formula reads
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
f ( z)  = L _ {\partial  D }  ^  \chi
 +
f ( z) - R _ {\partial  D }  ^  \chi
 +
\overline \partial \; f ( z) - B _ {D} \overline \partial \; f ( z) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818056.png" />.
+
where $  z \in D $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818057.png" /> is holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818058.png" />, then (a1) reduces to (*). Of particular importance are instances where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818059.png" />, and hence also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818060.png" />, is holomorphic as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818061.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818062.png" /> fixed — this can only occur if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818063.png" /> is pseudo-convex; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818064.png" /> is then a holomorphic support function (i.e. for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818065.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818067.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818068.png" /> is holomorphic in this neighbourhood and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818069.png" />), the existence of which is closely related to the existence of continuously varying holomorphic peaking functions. (A continuously varying holomorphic peaking function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818070.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818071.png" /> such that for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818072.png" />: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818073.png" /> is holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818074.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818075.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818077.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818078.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818080.png" /> is required to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818081.png" /> for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818082.png" />.) Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818083.png" /> is holomorphic for every continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818084.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818085.png" /> and the operator
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818086.png" /></td> </tr></table>
+
$$
 +
u  \mapsto  f  = - ( R _ {\partial  D }  ^  \chi  u + B _ {D} u )
 +
$$
  
 
solves the inhomogeneous Cauchy–Riemann equations
 
solves the inhomogeneous Cauchy–Riemann equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818087.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\left .
 +
\begin{array}{c}
 +
\overline \partial \; = u  \\
 +
\textrm{ with  integrability  condition  }  \overline \partial \; = 0 \\
 +
\end{array}
 +
\right \}
 +
$$
  
for continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818088.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818089.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818090.png" />. Formula (a1) can be generalized to give a representation formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818091.png" />-forms as well (see [[#References|[a2]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818092.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818094.png" /> depends on the domain only, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818095.png" /> denotes the Hölder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818096.png" />-norm and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058180/l05818097.png" /> 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.
+
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====
 
====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>
 
<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>

Revision as of 22:16, 5 June 2020


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
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