Representation via ∂-equation

Abstract

The Fefferman-Stein decomposition theorem for BMOA is to say that every BMOA-function f can be written as the sum f ₁∔f ₂ where f ₁, f ₂ ∈ H and Ref _j ∈ L∞(T). The main aim of this chapter is to extend this result to Q _p, p ∈ (0,1). This aim will be realized via: introducing Q _p(T), the non-holomorphic version of Q _p; finding the Q _p(T) ∩ L ^∞(T)-solutions to the ∂-equation; and presenting the Fefferman-Stein type decomposition for Q _p. As certain applications of the ∂-equation, we give the corona theorems for both Q _p ∩ H ^∞ and Q _p, and then show the interpolation theorem for Q _p ∩ H ^∞.