An operator-valued T(1) theorem for symmetric singular integrals in UMD spaces

TL;DR

This paper extends a classical operator-valued T(1) theorem to symmetric singular integrals in UMD spaces, providing new boundedness criteria that apply to non-commutative L^p spaces without additional Hardy space assumptions.

Contribution

It generalizes the T(1) theorem for symmetric singular integrals from Hilbert spaces to UMD spaces, including non-commutative L^p spaces, under natural BMO conditions.

Findings

Established boundedness of symmetric singular integrals in UMD spaces.

Extended results to non-commutative L^p spaces for all 1<p<∞.

Removed the need for Hardy space replacements in non-commutative settings.

Abstract

The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical ``$T(1)\in BMO$'' assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical $T(1)$ criterion still guarantees their $L^2$-boundedness on Hilbert space -valued functions. Here, these results are extended to general UMD (unconditional martingale differences) spaces with the same natural BMO condition for symmetrised paraproducts, and requiring in addition only the usual replacement of uniform bounds by $R$-bounds in the case of general singular integrals. In particular, under these assumptions, we obtain boundedness results on non-commutative $L^p$ spaces for all $1<p<\infty$, without the need to replace the domain or the target by a related non-commutative Hardy space as in the results of Hong et al. for $p\neq 2$.