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:

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]):

here, is the Cauchy–Fantappié differential form (see [a3])

where , , ,

c) Now, let be an -circular domain (a Reinhardt domain); then

where , all are non-negative integers, , .

If is a ball, then

where

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