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 $u:H^1\to L_1(\mu )$. Assume that on one hand $u$ admits an extension $u_1:L^1\to L_1(\mu )$ bounded with norm $C_1$, and on the other hand that $u$ admits an extension $u_{\infty }:L^{\infty }\to L_{\infty }(\mu )$ 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.