Schatten properties of commutators of fractional integrals on spaces of homogeneous type

TL;DR

This paper characterizes when commutators of fractional integral operators on spaces of homogeneous type belong to Schatten classes, extending classical Euclidean results to more general settings using kernel-based methods.

Contribution

It provides new Schatten class characterizations for commutators of fractional integrals on spaces of homogeneous type, covering operators beyond the Euclidean case with a kernel-based approach.

Findings

Schatten class membership characterized by Besov or fractional Sobolev space conditions.

Dichotomy in Schatten class membership depending on the parameter p and fractional order.

Extension of results to fractional Bessel operators and various dimensions.

Abstract

Extending classical results of Janson and Peetre (1988) on the Schatten class $S^p$ membership of commutators of Riesz potentials on the Euclidean space, we obtain analogous results for commutators $[b,T]$, where $T\in\{T_\varepsilon,\widetilde T_\alpha\}$ belongs to either one of two natural classes of fractional integral operators on a space of homogeneous type. Our approach is based on recent related work of Hyt\"{o}nen and Korte on singular (instead of fractional) integrals; working directly with the kernels, it differs from the Fourier analytic considerations of Janson and Peetre, covering new operators even when specialised to $\mathbb R^d$. The cleanest case of our characterization in spaces of lower dimension $d> 2$ and satisfying a $(1,2)$-Poincar\'e inequality is as follows. For a parameter $\varepsilon \in (0,\frac{1}{2}-\frac{1}{d})$ describing the order of the fractional integral $T_\varepsilon $, we have a dichotomy: If $\frac{d}{1+d\varepsilon }<p<\frac{1}{\varepsilon}$, then $[b,T_{\varepsilon}]\in S^p$ if and only if $b$ belongs to a suitable Besov (or fractional Sobolev) space. If $0<p\leq \frac{d}{1+d\varepsilon }$, then $[b,T_{\varepsilon}]\in S^p$ if and only if $b$ is constant. This is analogous to the result for singular integrals, where a similar cut-off happens at $p=d$, formally corresponding to fractional order $\varepsilon =0$. We also obtain results for other parameter values, including dimensions $0<d\leq 2$. As an application, these results are used to show Schatten properties of commutators of fractional Bessel operators, complementing recent related results of Fan, Lacey, Li, and Xiong (2025) on commutators of singular integrals in the Bessel setting.