Interpolation between $H^p$ Spaces and non-commutative generalizations, II

Abstract

We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the $\bar{\partial}$-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 $\bar{\partial}$-equation, which satisfies simultaneously a good $L_\infty$ estimate and a good $L_1$ estimate. This appears as a special case of our main result which can be stated as follows: Let $(\Omega, \mathcal A, \mu)$ be any measure space. Consider a bounded operator $u : H^1 \rightarrow L_1(\mu)$. Assume that on one hand $u$ admits an extension $u_1 : L^1 \rightarrow L_1(\mu)$ bounded with norm $C_1$, and on the other hand that $u$ admits an extension $u_\infty : L^\infty \rightarrow L_\infty(\mu)$ bounded with norm $C_\infty$. Then $u$ admits an extension $\tilde{u}$ which is bounded simultaneously from $L^1$ into $L_1(\mu)$ and from $L^\infty$ into $L_\infty(\mu)$ and satisfies

where $C$ is a numerical constant.