Spaces and Non-commutative Generalizations Ii*

We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the ¯ ∂-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution of the ¯ ∂-equation, which satisfies simultaneously a good L ∞ estimate and a good L 1 estimate. This appears as a special case of our main result which can be stated as follows: Let (Ω, A, µ) be any measure space. Consider a bounded operator u : H 1 → L 1 (µ). Assume that on one hand u admits an extension u 1 : L 1 → L 1 (µ) bounded with norm C 1 , and on the other hand that u admits an extension