On the Corona problem

Abstract

We show that if f1, f2 are bounded holomorphic functions in the unit ball\(\mathbb{B}\) of ℂn such that\(\forall z \in \mathbb{B}\), ¦f1(z)¦2 + ¦f2(z)2¦2 ≥ δ2 >; 0, then any functionh in the Hardy space\(H^p (\mathbb{B})\),p < +∞ can be decomposed ash = f1h1+ f2h2 with\(h_i \in H^p (\mathbb{B})\). The Corona theorem in\(\mathbb{B}\) would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.