Analytic continuation into a domain of a function given on part of the boundary
The following classical assertion is well known. Let be a simply connected bounded domain with smooth boundary
, and
. Then
![]() | (a1) |
if and only if extends into the domain
as a holomorphic function of the class
. For the multi-dimensional case
, instead of the form
, one takes an exterior differential form of class
.
If is defined only on a part of the boundary of
, then the existence of an analytic continuation into
cannot be decided by the vanishing of some family of continuous linear functionals as in (a1). Solutions to this problem were given from the 1950s onwards by many mathematicians, see, e.g. [a1], [a2].
Some very simple solutions are given below.
1) . Let
be the domain bounded by a part of the unit circle
and a smooth open arc
connecting two points of
and lying inside
. Let
. Set
![]() |
Then the following assertion holds: If , then there is a function
such that
if and only if
![]() | (a2) |
If is not identically zero, then (a2) is equivalent to
![]() |
2) . Let
be a
-circular convex domain in
, where
are natural numbers, i.e.,
implies
for
. In particular, for
this circular domain is a Cartan domain. Moreover, assume that
is convex and bounded and
. Furthermore, let
be the domain bounded by a part of
and a hyper-surface
dividing
into two parts and assume that the complement of
contains the origin. Consider the Cauchy–Fantappié differential form
![]() |
where ,
,
. Then
. By the Sard theorem,
for almost all
on
, where
is the homothetic transform of
. Assume that
on
and set
![]() |
![]() |
where ,
,
. Let
![]() |
where and
is the volume element in
. Here, all
and
are non-negative integers. Note that the integral moments
depend on
and
, but the moments
depend only on
.
The following assertion now holds: For a function to have an analytic continuation
with
, it is necessary and sufficient that the following two conditions are fulfilled:
i) is a
-function on
;
ii) .
A consequence of this is as follows. Let be a bounded convex
-circular domain (a Reinhardt domain). Set
. For a function
to have an analytic continuation in
as above it is necessary and sufficient that:
a) is a
-function on
;
b) .
References
[a1] | L. Aizenberg, "Carleman's formulas in complex analysis" , Kluwer Acad. Publ. (1993) |
[a2] | L. Aizenberg, "Carleman's formulas and conditions of analytic extendability" , Topics in Complex Analysis , Banach Centre Publ. , 31 , Banach Centre (1995) pp. 27–34 |
Analytic continuation into a domain of a function given on part of the boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_continuation_into_a_domain_of_a_function_given_on_part_of_the_boundary&oldid=15858