Noncommutative extrapolation theorems and applications

Abstract

In this paper, we prove some noncommutative analogues of Yano’s classical extrapolation theorem. Applying one of them to noncommutative martingales, we obtain a maximal inequality for noncommutative martingales from $L\log^2L$ to $L_1$. Moreover, the exponent $2$ is optimal. We also obtain the noncommutative analogue of the classical theorem of Burkholder and Chow on the iterations of two conditional expectations.