Multidimensional Analogs of Carleman Formulas with Integration over Boundary Sets of Maximal Dimension

Abstract

The initial idea of Carleman to construct a “quenching” function, making it possible to obtain from an integral representation of holomorphic functions involving integration over the whole boundary ∂D of a domain D an integral representation involving integration over a set M ⊂ ∂D, rests on the availability of a function φ(z) of class H^∞(D) satisfying two conditions (see sec. 1):

$$ 1)|\phi (\varsigma )| = 1{\rm{almost}}{\rm{everywhere}}{\rm{on}}\partial D\backslash M, $$

$$ 2)|\phi (\varsigma )|{\rm{ > }}1{\rm{}}D. $$