Endpoint Schatten class properties of commutators

2024/05/17 by Rupert L. Frank, Fedor Sukochev, Frank, Rupert L. +3 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2405.10652

openalex publication_date 2024/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study trace ideal properties of the commutators [(-Δ)^\fracε2,M_f] of a power of the Laplacian with the multiplication operator by a function f on \mathbb R^d. For a certain range of ε∈\mathbb R, we show that this commutator belongs to the weak Schatten class \mathcal L_\frac d1-ε,∞ if and only if the distributional gradient of f belongs to L_\frac d1-ε. Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals.

Cited by

• Singular value asymptotics on compact smooth Riemaniann manifolds
• Schatten properties of commutators of fractional integrals on spaces of homogeneous type

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