Lorentz-Schatten Thresholds for Commutators on Metric Spaces

Abstract

We develop a refined, quantitative theory for Schatten properties of commutators \([b,T]:f\mapsto b(Tf)-T(bf)\) of Calderón-Zygmund type operators on (quasi-)metric measure spaces. Building on the dyadic representation framework and oscillation characterizations, we introduce a scale of \emph{Lorentz--oscillation} functionals that capture the distribution of first-order oscillations of \(b\) across space and scale. These yield sharp two-sided criteria for the inclusion \([b,T]\in S_{p,q}\) in terms of natural Lorentz refinements of the Besov/Sobolev data of \(b\). In particular, at the critical index \(p=d\) (the lower dimension), we identify novel borderline phenomena: while \(S_{d,\infty}\) corresponds to first-order Sobolev regularity, intermediate ideals \(S_{d,q}\) (\(1\le q<\infty\)) are shown to be equivalent to strengthened \emph{logarithmic} and \emph{Lorentz} Sobolev-type conditions on \(b\). For \(p>d\) we obtain norm equivalences \(\|[b,T]\|_{S_{p,q}}\simeq \|b\|_{\mathcal{B}^{d/p}_{p,q}}\) on general spaces of homogeneous type, with weighted \(A_2\) extensions, and we delineate the precise failure for \(p\le d\) (strong classes) beyond constants. The methods unify lower bounds via approximate weak factorization and Carleson embeddings with upper bounds via dyadic shift commutator analysis and real interpolation of ideals. Illustrations are given on Ahlfors regular spaces, Heisenberg/Carnot groups, and the Bessel setting.