The sheaf of H^p-functions in product domains

Sergio Eduardo Zarantonello

Vol. 93 (1981), No. 2, 479–493

DOI: 10.2140/pjm.1981.93.479

Abstract

Let W = W₁ ×W₂ ×⋯×W_n be a bounded polydomain in Cⁿ such that the boundary of each W_i consists of finitely many disjoint Jordan curves. The correspondence that assigns to every relatively open polydomain V in W (the closure of W) the Hardy space ℋ^p(V ∩W), defines a sheaf ℋ_W^p over W. This sheaf is locally determined in the sense that Γ(W,ℋ_W^p) is canonically isomorphic to ℋ^p(W). In this paper we prove, for any 0 < p < ∞ and all integers q ≥ 1, that the cohomology groups H^q(W,ℋ_W^p) are trivial.

Mathematical Subject Classification 2000

Primary: 32A35

Secondary: 32C35, 46J15

Milestones

Received: 19 June 1978

Revised: 30 October 1978

Published: 1 April 1981