On a class of holomorphic functions representable by Carleman formulas in some class of bounded, simply connected domains from their values on an analytic arc

Let sript U sign be a bounded, simply connected domain with Jordan rectifiable boundary and let M ⊂ ∂ sript U sign be an open analytic arc whose Lebesgue measure satisfies 0 < m(M) < m(∂ sript U sign. Our result gives a complete description of the class of holomorphic functions in sript U sign which are represented by the Carleman formulas on the open arc M, when ∂ sript U sign is almost regular with respect to M (Section 2). That is, we give a type of integral representation formulas for functions holomorphic in a domain sript U sign by its values on a part M of the boundary ∂ sript U sign. This class is denoted by sript N sign ℋ¹ _M(sript U sign).