The Hp corona theorem in analytic polyhedra

Jörgen Boo

doi:10.1007/BF02559968

October 1997 The H^p corona theorem in analytic polyhedra

Jörgen Boo

Author Affiliations +

Ark. Mat. 35(2): 225-251 (October 1997). DOI: 10.1007/BF02559968

Abstract

TheH^p corona problem is the following: Let g₁, ..., g_m be bounded holomorphic functions with 0<δ≤Σ‖g_i‖. Can we, for any H^p function ϕ, find H^p functions u₁, ..., u_m such that Σg_iu_i=ϕ? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. The H^p estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.

Note

I am very grateful to my advisor, Mats Andersson, for proposing the subject of this paper and for showing great interest in the project. I also want to express my thanks to Hasse Carlsson and Joaquim Ortega Cerdà for many helpful discussions. Finally, I wish to thank the referee for several comments which helped to improve the exposition.