Embeddings between operator-valued dyadic BMO spaces

TL;DR

This paper explores the structure of operator-valued dyadic BMO spaces, comparing different characterizations and operator space structures, and provides sharp estimates for function sweeps and bilinear extensions.

Contribution

It introduces new operator space structures on dyadic BMO spaces and establishes sharp dimensional growth estimates for related functions.

Findings

Different operator space structures are identified on dyadic BMO spaces.

Sharp dimensional growth estimates are obtained for the sweep of functions.

Results extend scalar BMO characterizations to the operator-valued setting.

Abstract

We investigate a scale of dyadic operator-valued BMO spaces, corresponding to the different yet equivalent characterizations of dyadic BMO in the scalar case. In the language of operator spaces, we investigate different operator space structures on the scalar dyadic BMO space which arise naturally from the different characterisations of scalar BMO. We also give sharp dimensional growth estimates for the sweep of functions and its bilinear extension in some of those different dyadic BMO spaces.