Noncommutative Stein implication for almost-everywhere convergence

Establish a general noncommutative analogue of Stein’s implication that almost-everywhere convergence of a sequence of convolution operators on L_p yields a weak-type maximal estimate, extending beyond the weaker form proved for 1<p≤2 in the paper.

Background

The paper develops a noncommutative version of Stein’s extrapolation principle sufficient for the polynomial ergodic averages under study. The result obtained is a weaker form of the classical implication and is proved using a noncommutative replacement for the Borel–Cantelli argument.

The authors explain that a general analogue is substantially more difficult because Stein’s proof relies on pointwise maximal functions and point-dependent techniques, including the Borel–Cantelli argument, which do not directly exist in the noncommutative setting. The appendix provides an L1 result in the spirit of Burkholder’s theorem, but does not resolve the general problem.