Namespaces
Variants
Actions

Difference between revisions of "Bochner-Martinelli representation formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (typo)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
 +
<!--
 +
b0167201.png
 +
$#A+1 = 55 n = 0
 +
$#C+1 = 55 : ~/encyclopedia/old_files/data/B016/B.0106720 Bochner\ANDMartinelli representation formula,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''Bochner–Martinelli representation, Bochner–Martinelli formula''
 
''Bochner–Martinelli representation, Bochner–Martinelli formula''
  
An integral representation for holomorphic functions, which is defined as follows [[#References|[1]]], [[#References|[2]]]. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167201.png" /> be holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167202.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167203.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167204.png" /> be continuous in its closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167205.png" />. Then the expression
+
An integral representation for holomorphic functions, which is defined as follows [[#References|[1]]], [[#References|[2]]]. Let the function $  f $
 +
be holomorphic in a domain $  D \subset  {\mathbf C  ^ {n} } $
 +
with piecewise-smooth boundary $  \partial  D $,  
 +
and let $  f $
 +
be continuous in its closure $  \overline{D}\; $.  
 +
Then the expression
 +
 
 +
$$ \tag{1 }
 +
 
 +
\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/b/b016/b016720/b0167206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\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/b/b016/b016720/b0167207.png" /></td> </tr></table>
+
\frac{f( \zeta ) }{| \zeta -z |  ^ {2n} }
 +
\times
 +
$$
  
<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/b/b016/b016720/b0167208.png" /></td> </tr></table>
+
$$
 +
\times
 +
\sum _ { j=1 } ^ { n }  ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} ) \
 +
d \overline \zeta \; _ {1}  \wedge  d \zeta _ {1}  \wedge \dots \wedge \
 +
[d \overline \zeta \; _ {j} ]  \wedge  d \zeta _ {j}  \wedge \dots
 +
\wedge  d \zeta _ {n\ } =
 +
$$
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b0167209.png" /> means that the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672010.png" /> is to be omitted, is called the Bochner–Martinelli representation. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672011.png" /> this representation is identical with the Cauchy integral formula (cf. [[Cauchy integral|Cauchy integral]]), but for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672012.png" /> its kernel is not holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672013.png" />, which is the reason for the limited applicability of the Bochner–Martinelli representation in the theory of functions of several complex variables. The kernel of the Bochner–Martinelli representation is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672014.png" />-differential form of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672015.png" />:
+
$$
 +
= \
 +
\left \{
 +
\begin{array}{cl}
 +
f(z), & \textrm{ if }
 +
z \in D,  \\
 +
0, & \textrm{ if }  z \notin \overline{D}\; ,  \\
 +
\end{array}
  
<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/b/b016/b016720/b01672016.png" /></td> </tr></table>
+
\right .$$
  
<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/b/b016/b016720/b01672017.png" /></td> </tr></table>
+
here  $  [d \overline \zeta \; _ {j} ] $
 +
means that the term  $  d \overline \zeta \; _ {j} $
 +
is to be omitted, is called the Bochner–Martinelli representation. For  $  n = 1 $
 +
this representation is identical with the Cauchy integral formula (cf. [[Cauchy integral|Cauchy integral]]), but for  $  n > 1 $
 +
its kernel is not holomorphic in  $  z $,
 +
which is the reason for the limited applicability of the Bochner–Martinelli representation in the theory of functions of several complex variables. The kernel of the Bochner–Martinelli representation is the  $  \zeta $-
 +
differential form of bidegree  $  (n, n - 1) $:
  
which is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672018.png" />, has a singular point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672019.png" />, and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672021.png" />-closed (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672022.png" />) outside the singular point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672023.png" />, the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672024.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672025.png" />, where
+
$$
 +
\omega ( \zeta , z )  = \
 +
(n-1)! over {(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/b/b016/b016720/b01672026.png" /></td> </tr></table>
+
\frac{1}{| \zeta -z |  ^ {2n} }
 +
\times
 +
$$
  
is a form of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672027.png" />, the coefficient of which is a fundamental solution of the Laplace equation; here
+
$$
 +
\times
 +
\sum _ { j=1 } ^ { n }  ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} )
 +
d \overline \zeta \; _ {1}  \wedge  d \zeta _ {1}  \wedge \dots
 +
\wedge  [ d \overline \zeta \; _ {j} ]  \wedge
 +
d \zeta _ {j}  \wedge \dots \wedge  d \overline \zeta \; _ {n}  \wedge  d \zeta _ {n} ,
 +
$$
  
<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/b/b016/b016720/b01672028.png" /></td> </tr></table>
+
which is defined in  $  \mathbf C  ^ {n} $,
 +
has a singular point at  $  \zeta = z $,
 +
and is  $  \overline \partial \; $-
 +
closed (i.e.  $  \overline \partial \; \omega = 0 $)
 +
outside the singular point. If  $  n > 1 $,
 +
the form  $  \omega $
 +
is equal to  $  \partial  \omega  ^  \prime  ( \zeta , z) $,
 +
where
  
The following integral representation, which generalizes formula (1), is the analogue of the Cauchy–Green formula (cf. [[Cauchy integral|Cauchy integral]]): If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672029.png" /> is continuously differentiable in the closure of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672030.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672031.png" />, then, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672032.png" />,
+
$$
 +
\omega  ^  \prime  ( \zeta , z)  =  -
 +
(n-2)! over {(2 \pi i) ^ {n} } \cdot
  
<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/b/b016/b016720/b01672033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{1}{| \zeta -z |  ^ {2n-2} }
 +
 
 +
\sum _ { j=1 } ^ { n }
 +
\left ( \prod _ { k\neq j }
 +
d \overline \zeta \; _ {k}  \wedge  d \zeta _ {k} \right )
 +
$$
 +
 
 +
is a form of bidegree  $  (n - 1, n - 1) $,
 +
the coefficient of which is a fundamental solution of the Laplace equation; here
 +
 
 +
$$
 +
\partial  \phi  = \
 +
\sum dz _ {k}
 +
\frac{\partial  \phi }{\partial  z _ {k} }
 +
\
 +
\textrm{ and } \ \
 +
\overline \partial \; \phi  = \
 +
\sum d \overline{z}\; _ {k}
 +
\frac{\partial  \phi }{\partial  z _ {k} }
 +
.
 +
$$
 +
 
 +
The following integral representation, which generalizes formula (1), is the analogue of the Cauchy–Green formula (cf. [[Cauchy integral|Cauchy integral]]): If the function  $  f $
 +
is continuously differentiable in the closure of a domain  $  D \subset  \mathbf C  ^ {n} $
 +
with piecewise-smooth boundary  $  \partial  D $,
 +
then, for any point  $  z \in D $,
 +
 
 +
$$ \tag{2 }
 +
f(z)  = \int\limits _ {\partial  D }
 +
f( \zeta ) \omega ( \zeta , z) -
 +
\int\limits _ { D } \overline \partial \; f( \zeta )
 +
\wedge \omega ( \zeta , z).
 +
$$
  
 
The function
 
The function
  
<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/b/b016/b016720/b01672034.png" /></td> </tr></table>
+
$$
 +
\widehat{f}  (z)  = \
 +
\int\limits _  \Gamma  f( \zeta )
 +
\omega ( \zeta , z),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672035.png" /> is a smooth hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672037.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672038.png" /> which is Lebesgue-integrable, is said to be an integral of Bochner–Martinelli type. As for Cauchy-type integrals, Sokhotskii's formula, with the usual restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672040.png" />, is applicable to Bochner–Martinelli-type integrals. A Bochner–Martinelli-type integral is a complex function which is harmonic everywhere outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672041.png" />; in the general case this function is holomorphic only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672042.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672043.png" />, then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672044.png" />, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672045.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672046.png" /> is equivalent to the holomorphy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672047.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672048.png" />.
+
where $  \Gamma $
 +
is a smooth hypersurface in $  \mathbf R  ^ {2n} = \mathbf C  ^ {n} $
 +
and $  f $
 +
is a function on $  \Gamma $
 +
which is Lebesgue-integrable, is said to be an integral of Bochner–Martinelli type. As for Cauchy-type integrals, Sokhotskii's formula, with the usual restrictions on $  \Gamma $
 +
and $  f $,  
 +
is applicable to Bochner–Martinelli-type integrals. A Bochner–Martinelli-type integral is a complex function which is harmonic everywhere outside $  \Gamma $;  
 +
in the general case this function is holomorphic only for $  n = 1 $.  
 +
If $  \Gamma = \partial  D $,  
 +
then if $  n \geq  1 $,  
 +
the condition $  \widehat{f}  (z) \equiv 0 $
 +
outside $  \overline{D}\; $
 +
is equivalent to the holomorphy of $  \widehat{f}  $
 +
in $  D $.
  
 
The Bochner–Martinelli representation is employed to demonstrate other integral representations (e.g. the [[Bergman–Weil representation|Bergman–Weil representation]]), in holomorphic continuation from the boundary, and also in the theory of boundary values of holomorphic functions of several complex variables. It was introduced by S. Bochner [[#References|[1]]] and by E. Martinelli [[#References|[2]]].
 
The Bochner–Martinelli representation is employed to demonstrate other integral representations (e.g. the [[Bergman–Weil representation|Bergman–Weil representation]]), in holomorphic continuation from the boundary, and also in the theory of boundary values of holomorphic functions of several complex variables. It was introduced by S. Bochner [[#References|[1]]] and by E. Martinelli [[#References|[2]]].
Line 37: Line 147:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Bochner,  "Analytic and meromorphic continuation by means of Green's formula"  ''Ann. of Math. (2)'' , '''44''' :  4  (1943)  pp. 652–673</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Martinelli,  ''Rend. Accad. Ital.'' , '''9'''  (1938)  pp. 269–283</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Bochner,  "Analytic and meromorphic continuation by means of Green's formula"  ''Ann. of Math. (2)'' , '''44''' :  4  (1943)  pp. 652–673</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Martinelli,  ''Rend. Accad. Ital.'' , '''9'''  (1938)  pp. 269–283</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Sokhotskii's formula is also known as Plemelj's formula. The Cauchy–Green formula is also known as the Cauchy–Pompeiu formula.
 
Sokhotskii's formula is also known as Plemelj's formula. The Cauchy–Green formula is also known as the Cauchy–Pompeiu formula.
  
The Bochner–Martinelli kernel is a special case of a Cauchy–Fantappié kernel. The integral representation (2) solves the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672050.png" />-equation:
+
The Bochner–Martinelli kernel is a special case of a Cauchy–Fantappié kernel. The integral representation (2) solves the $  \overline \partial \; $-
 +
equation:
  
<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/b/b016/b016720/b01672051.png" /></td> </tr></table>
+
$$
 +
\overline \partial \; = \
 +
v \  \textrm{ with } \ \
 +
\overline \partial \; = 0,
 +
$$
  
for compactly-supported <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672052.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672053.png" />, by substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672054.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672055.png" /> in the right-hand side of (2) and omitting the integral over the boundary. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672056.png" /> is not compactly supported, the boundary integral causes difficulties. These can be solved for strictly pseudo-convex domains and the Bochner–Martinelli kernel then occurs in an explicit solution operator for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016720/b01672057.png" />-equation.
+
for compactly-supported $  (0, 1) $-
 +
forms $  v $,  
 +
by substituting $  v $
 +
for $  \partial  f $
 +
in the right-hand side of (2) and omitting the integral over the boundary. When $  v $
 +
is not compactly supported, the boundary integral causes difficulties. These can be solved for strictly pseudo-convex domains and the Bochner–Martinelli kernel then occurs in an explicit solution operator for the $  \overline \partial \; $-
 +
equation.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,  "Function theory of several complex variables" , Wiley  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. [G.M. Khenkin] Henkin,  J. Leiterer,  "Theory of functions on complex manifolds" , Birkhäuser  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.M. Range,  "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. 1, Sect. 3</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,  "Function theory of several complex variables" , Wiley  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. [G.M. Khenkin] Henkin,  J. Leiterer,  "Theory of functions on complex manifolds" , Birkhäuser  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.M. Range,  "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. 1, Sect. 3</TD></TR></table>

Latest revision as of 19:50, 29 May 2020


Bochner–Martinelli representation, Bochner–Martinelli formula

An integral representation for holomorphic functions, which is defined as follows [1], [2]. Let the function $ f $ be holomorphic in a domain $ D \subset {\mathbf C ^ {n} } $ with piecewise-smooth boundary $ \partial D $, and let $ f $ be continuous in its closure $ \overline{D}\; $. Then the expression

$$ \tag{1 } \frac{(n-1)! }{(2 \pi i) ^ {n} } \int\limits _ {\partial D } \frac{f( \zeta ) }{| \zeta -z | ^ {2n} } \times $$

$$ \times \sum _ { j=1 } ^ { n } ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} ) \ d \overline \zeta \; _ {1} \wedge d \zeta _ {1} \wedge \dots \wedge \ [d \overline \zeta \; _ {j} ] \wedge d \zeta _ {j} \wedge \dots \wedge d \zeta _ {n\ } = $$

$$ = \ \left \{ \begin{array}{cl} f(z), & \textrm{ if } z \in D, \\ 0, & \textrm{ if } z \notin \overline{D}\; , \\ \end{array} \right .$$

here $ [d \overline \zeta \; _ {j} ] $ means that the term $ d \overline \zeta \; _ {j} $ is to be omitted, is called the Bochner–Martinelli representation. For $ n = 1 $ this representation is identical with the Cauchy integral formula (cf. Cauchy integral), but for $ n > 1 $ its kernel is not holomorphic in $ z $, which is the reason for the limited applicability of the Bochner–Martinelli representation in the theory of functions of several complex variables. The kernel of the Bochner–Martinelli representation is the $ \zeta $- differential form of bidegree $ (n, n - 1) $:

$$ \omega ( \zeta , z ) = \ (n-1)! over {(2 \pi i) ^ {n} } \frac{1}{| \zeta -z | ^ {2n} } \times $$

$$ \times \sum _ { j=1 } ^ { n } ( \overline \zeta \; _ {j} - \overline{z}\; _ {j} ) d \overline \zeta \; _ {1} \wedge d \zeta _ {1} \wedge \dots \wedge [ d \overline \zeta \; _ {j} ] \wedge d \zeta _ {j} \wedge \dots \wedge d \overline \zeta \; _ {n} \wedge d \zeta _ {n} , $$

which is defined in $ \mathbf C ^ {n} $, has a singular point at $ \zeta = z $, and is $ \overline \partial \; $- closed (i.e. $ \overline \partial \; \omega = 0 $) outside the singular point. If $ n > 1 $, the form $ \omega $ is equal to $ \partial \omega ^ \prime ( \zeta , z) $, where

$$ \omega ^ \prime ( \zeta , z) = - (n-2)! over {(2 \pi i) ^ {n} } \cdot \frac{1}{| \zeta -z | ^ {2n-2} } \sum _ { j=1 } ^ { n } \left ( \prod _ { k\neq j } d \overline \zeta \; _ {k} \wedge d \zeta _ {k} \right ) $$

is a form of bidegree $ (n - 1, n - 1) $, the coefficient of which is a fundamental solution of the Laplace equation; here

$$ \partial \phi = \ \sum dz _ {k} \frac{\partial \phi }{\partial z _ {k} } \ \textrm{ and } \ \ \overline \partial \; \phi = \ \sum d \overline{z}\; _ {k} \frac{\partial \phi }{\partial z _ {k} } . $$

The following integral representation, which generalizes formula (1), is the analogue of the Cauchy–Green formula (cf. Cauchy integral): If the function $ f $ is continuously differentiable in the closure of a domain $ D \subset \mathbf C ^ {n} $ with piecewise-smooth boundary $ \partial D $, then, for any point $ z \in D $,

$$ \tag{2 } f(z) = \int\limits _ {\partial D } f( \zeta ) \omega ( \zeta , z) - \int\limits _ { D } \overline \partial \; f( \zeta ) \wedge \omega ( \zeta , z). $$

The function

$$ \widehat{f} (z) = \ \int\limits _ \Gamma f( \zeta ) \omega ( \zeta , z), $$

where $ \Gamma $ is a smooth hypersurface in $ \mathbf R ^ {2n} = \mathbf C ^ {n} $ and $ f $ is a function on $ \Gamma $ which is Lebesgue-integrable, is said to be an integral of Bochner–Martinelli type. As for Cauchy-type integrals, Sokhotskii's formula, with the usual restrictions on $ \Gamma $ and $ f $, is applicable to Bochner–Martinelli-type integrals. A Bochner–Martinelli-type integral is a complex function which is harmonic everywhere outside $ \Gamma $; in the general case this function is holomorphic only for $ n = 1 $. If $ \Gamma = \partial D $, then if $ n \geq 1 $, the condition $ \widehat{f} (z) \equiv 0 $ outside $ \overline{D}\; $ is equivalent to the holomorphy of $ \widehat{f} $ in $ D $.

The Bochner–Martinelli representation is employed to demonstrate other integral representations (e.g. the Bergman–Weil representation), in holomorphic continuation from the boundary, and also in the theory of boundary values of holomorphic functions of several complex variables. It was introduced by S. Bochner [1] and by E. Martinelli [2].

References

[1] S. Bochner, "Analytic and meromorphic continuation by means of Green's formula" Ann. of Math. (2) , 44 : 4 (1943) pp. 652–673
[2] E. Martinelli, Rend. Accad. Ital. , 9 (1938) pp. 269–283
[3] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

Comments

Sokhotskii's formula is also known as Plemelj's formula. The Cauchy–Green formula is also known as the Cauchy–Pompeiu formula.

The Bochner–Martinelli kernel is a special case of a Cauchy–Fantappié kernel. The integral representation (2) solves the $ \overline \partial \; $- equation:

$$ \overline \partial \; f = \ v \ \textrm{ with } \ \ \overline \partial \; v = 0, $$

for compactly-supported $ (0, 1) $- forms $ v $, by substituting $ v $ for $ \partial f $ in the right-hand side of (2) and omitting the integral over the boundary. When $ v $ is not compactly supported, the boundary integral causes difficulties. These can be solved for strictly pseudo-convex domains and the Bochner–Martinelli kernel then occurs in an explicit solution operator for the $ \overline \partial \; $- equation.

References

[a1] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)
[a2] G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984)
[a3] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 1, Sect. 3
How to Cite This Entry:
Bochner-Martinelli representation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner-Martinelli_representation_formula&oldid=22145
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article