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

2025/11/04 by Hytönen, Tuomas, Wu, Lin
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.2511.02686

openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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∈\T_ε,\widetilde T_α\ 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 Hytönen 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)-Poincaré inequality is as follows. For a parameter ε ∈ (0,(1)/(2)-(1)/(d)) describing the order of the fractional integral T_ε , we have a dichotomy: If (d)/(1+dε )

Citations

• Characterisations of Sobolev spaces and constant functions over metric spaces
• Schatten properties of commutators on metric spaces
• Complex median method and Schatten class membership of commutators
• Endpoint Schatten class properties of commutators
• Schatten--Lorentz characterization of Riesz transform commutator associated with Bessel operators
• The Riesz Transform and Fractional Integral Operators in the Bessel Setting
• Solving a Dirichlet problem for unbounded domains via a conformal transformation
• The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator
• The Poincaré inequality is an open ended condition
• Nearly weakly orthonormal sequences, singular value estimates, and Calderon-Zygmund operators
• On the parabolic kernel of the Schrödinger operator

Related

• Schatten properties of commutators on metric spaces
• Commutators for certain fractional type operators on weighted spaces and Orlicz-Morrey spaces
• Fractional Bloom boundedness of commutators in spaces of homogeneous type
• A short proof of commutator estimates
• Estimates for commutators of fractional differential operators via harmonic extension