Interpolation between Hp Spaces and non-commutative generalizations

We continue an investigation started in a preceding paper. We discuss tha classical result 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 for the ?-equation, which satisfies simultaneously a good L8 estimate and a good L1 estimate. This appears as a special case of our main result which can be stated as follows: Let (O, A, µ) be any measure space. Consider a bounded operator u: H1 ? L1(µ). Assume that on one hand u admits an extension u1: L1 ? L1(µ) bounded with norm C1, and on the other hand that u admits an extension u8: L8 ? L8(µ) bounded with norm C8. Then u admits an extension u' which is bounded simultaneously from L1 into L1(µ) and from L8 into L8(µ) and satisfies ||u': L8 ? L8(µ)|| = C C8 ||u': L1 ? L1(µ)|| = C C1 where C is a numerical constant