Multidimensional Carleman Formulas for Sets of Smaller Dimension

Abstract

We generalize the Carleman formula (1.3) to functions which are holomorphic in the polydisk U^j = U(0, 1) = {z: z ∈ ℂ^j ,|Z _i | < 1, i = 1,...,j} denote the distinguished boundary of U^j by Δ^j. We write a variable z ∈ ℂ^j as z^(j), and denote by H ¹ _j = H ¹ _j (U^j )the Hardy class of functions in A(U^j) for which

$$ \int\limits_{{\Delta ^j}} {\left| {f(r{\varsigma ^{(j)}}} \right|} {\rm{ }}d{m_j}{\rm{ < }}C. $$