Carleman formulas
Let be a bounded domain in
with piecewise smooth boundary
, and let
be a set of positive
-dimensional Lebesgue measure in
.
The following boundary value problem can then be posed (cf. also Boundary value problems of analytic function theory): Given a holomorphic function in
that is sufficiently well-behaved up to the boundary
(for example,
is continuous in
,
, or
belongs to the Hardy class
) how can it be reconstructed inside
by its values on
by means of an integral formula?
Three methods of solution are known, due to:
1) Carleman–Goluzin–Krylov;
2) M.M. Lavrent'ev; and
3) A.M. Kytmanov. See [a1].
The following are some very simple solutions:
a) . If
is a smooth arc connecting two points of the unit circle
and lying inside
and
is the domain bounded by a part of
and the arc
, with
, then for
and
the following Carleman formula holds:
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b) . Let
be a circular convex bounded domain (a Cartan domain) with
-boundary and let
be a piecewise smooth hypersurface intersecting
and cutting from it the domain
, with
. Then there exists a Cauchy–Fantappié formula for the domain
with kernel holomorphic in
. Let
,
, and
. Assume that there exists a vector-valued function (a "barrier" )
,
,
, such that
,
, and
smoothly extends to
on
, where
. Then for every function
and
, the following Carleman formula with holomorphic kernel is valid (see [a2]):
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here, is the Cauchy–Fantappié differential form (see [a3])
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where ,
,
,
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c) Now, let be an
-circular domain (a Reinhardt domain); then
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where , all
are non-negative integers,
,
.
If is a ball, then
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where
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In all the above Carleman formulas the limits are understood in the sense of uniform convergence on compact subsets of . A description of the class of holomorphic functions representable by Carleman formulas is given in [a4]. In [a1] applications of Carleman formulas in analysis and in mathematical physics can be found as well.
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 |
[a3] | L. Aizenberg, A.P. Yuzhakov, "Integral representation and residues in multidimensional complex analysis" , Amer. Math. Soc. (1983) (In Russian) |
[a4] | L. Aizenberg, A. Tumanov, A. Vidras, "The class of holomorphic functions representable by Carleman formula" Ann. Scuola Norm. Pisa , 27 : 1 (1998) pp. 93–105 |
Carleman formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_formulas&oldid=50276