The Cauchy kernel for cones
A new representation of the Cauchy kernel KΓ for an arbitrary acute convex cone Γ in R is found. The domain of holomorphy ofKΓ is described. An estimation of the growth of KΓ near the singularities is given. Introduction. The n-dimensional Cauchy kernel is a holomorphic function on (C \ {0})n defined by (C \ {0}) 3 (z1, . . . , zn) 7→ 1 z1 . . . zn . This function can be regarded as the Laplace transform of the characteristic function of R+: 1 z1 . . . zn = (−1) R+ e〈y,z〉 dy for −Re z ∈ R+ := (R+). Generalizing this formula, S. Bochner defined in [1] the Cauchy kernel KΓ for an arbitrary cone Γ ⊂ Rn by KΓ (z) := (−1) Γ ∗ e〈y,z〉 dy for Re z ∈ − Int(Γ ∗), (0.1) where Γ ∗ is the dual cone to Γ . V. Vladimirov found and studied in [4]–[6] another representation of the Cauchy kernel, KΓ (z) = (n− 1)! Γ ∗∩Sn−1 dσ 〈z, σ〉n , (0.2) and he proved that KΓ has a holomorphic continuation to the set ΩΓ := ⋂ σ∈Γ ∗∩Sn−1 {z : 〈z, σ〉 6= 0}. 2000 Mathematics Subject Classification: 32A26, 32D15.