$A_\infty$-invariance of oscillatory norms, and Schatten characterisations of commutators

Abstract

Schatten class properties of commutators $[b,T]$ of pointwise multipliers $b$ and singular integral operators $T$ have been characterised in a variety of settings. An abstract framework, covering many of these results as special cases, was proposed by the author [arXiv:2411.02613]. However, recent results about commutators of the concrete Bessel-Riesz transforms by Fan-Li-Sukochev-Zanin [arXiv:2411.14928] are beyond this abstract setting. In this work, we present an extension of the framework of [arXiv:2411.02613], introducing two measures $\mu$ and $\nu$ that are $A_\infty$-equivalent to each other. The commutators act on a given space $L^2(\mu)$, but the characterising function space norms of the multiplier $b$ are taken with respect to another measure $\nu$. In this way, assumptions like Ahlfors regularity and Poincar\'e inequality on the original measure $\mu$ may be relaxed, as long as there is an $A_\infty$-equivalent measure $\nu$ that satisfies these assumptions. In the Bessel example, the original $\mu$ fails to be Ahlfors regular, but $\nu$ is simply the Lebesgue measure. Within this framework, the Schatten norm characterisations of commutators of the Bessel-Riesz transforms at the critical-index by Fan-Li-Sukochev-Zanin [op cit.] are recovered by a completely different argument, replacing non-commutative techniques by real-variable harmonic analysis and hardly using any specifics of the Bessel setting. As a by-product, we also obtain a simpler characterisation in the non-critical case, replacing an ad-hoc Besov space of Fan-Lacey-Li-Xiong [J. Funct. Anal. 2026] by a classical Besov space.