Schatten properties of commutators on metric spaces

We characterise the Schatten class $S^p$ properties of commutators $[b,T]$ of singular integrals and pointwise multipliers in a general framework of (quasi-)metric measure spaces. This covers, unifies, and extends a range of previous results in different special cases. As in the classical results on $\mathbb R^d$, the characterisation has three parts: (1) For $p>d$, we have $[b,T]\in S^p$ if and only if $b$ is in a suitable Besov (or fractional Sobolev) space. (2) For $p\leq d$, we have $[b,T]\in S^p$ if and only if $b$ is constant. (3) For $p=d$, we have $[b,T]\in S^{d,\infty}$ (a weak-type Schatten class) if and only if $b$ is in a first-order Sobolev space. Result (1) extends to all spaces of homogeneous type as long as there are appropriate singular integrals, but for the more delicate properties (2) and (3), we assume a complete doubling metric space supporting a suitable Poincar\'e inequality, which is still very general. These latter results depend on new characterisations of constant functions and Sobolev spaces over such spaces obtained in a companion paper of the author with R. Korte. Even when specialised to various concrete domains considered earlier, the present results extend ones available in the literature by covering a larger class of operators with minimal kernel assumptions, removing a-priori assumptions on the pointwise multiplier $b$, and allowing Schatten classes on the weighted spaces $L^2(w)$ with an arbitrary Muckenhoupt weight $w\in A_2$. Even on $\mathbb R^d$, such weighted results were previously known for a few special operators $T$ only, and on all other domains, they are completely new.