Martingale inequalities and Operator space structures on $L_p$

TL;DR

This paper introduces a new operator space structure on $L_p$ for even integers, simplifying key inequalities and linking to the operator Hilbert space, with extensions to non-commutative $L_p$ spaces.

Contribution

The paper presents a novel operator space structure on $L_p$ for even $p$, providing more natural forms of classical inequalities and extending to non-commutative spaces.

Findings

The span of Rademacher functions is completely isomorphic to the operator Hilbert space $OH$.

The square function of a martingale difference sequence has a simplified form.

The new structure extends to non-commutative $L_p$ spaces with similar properties.

Abstract

We describe a new operator space structure on $L_p$ when $p$ is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's martingale inequalities have a very natural form:\ the span of the Rademacher functions is completely isomorphic to the operator Hilbert space $OH$, and the square function of a martingale difference sequence $d_n$ is $\Sigma \ d_n\otimes \bar d_n$. Various inequalities from harmonic analysis are also considered in the same operator valued framework. Moreover, the new operator space structure also makes sense for non commutative $L_p$-spaces with analogous results.