# 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) .