Characterisations of Sobolev spaces and constant functions over metric spaces

2025/08/11 by Tuomas Hytönen, Hytönen, Tuomas, Riikka Korte +1 · 1 citation
Mathematics · #42B35 #46E36 #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2508.07801

openalex publication_date 2025/08/11 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

In a doubling metric measure space (X,ρ,μ) supporting a Poincaré inequality, we give a new characterisation of first-order Sobolev spaces by mean oscillations, and extend previous characterisations of constant functions in terms of the finiteness of certain integrals through a new approach. As a key tool of independent potential, we introduce a novel ``macroscopic'' Poincaré inequality, whose right-hand side has oscillations of the same form as the left-hand side, but at a smaller macroscopic scale r∈(0,R). Besides intrinsic interest, these results are motivated by applications to quantitative compactness properties of commutators [f,T] of pointwise multipliers and singular integrals. With pivotal use of the present results, a characterisation of commutator mapping properties, over the same class of general domains (X,ρ,μ), is obtained in a companion paper.

Citations

• Characterizations of the Sobolev norms and the total variation via nonlocal functionals, and related problems
• A new characterization of Sobolev spaces on Lipschitz differentiability spaces
• Schatten properties of commutators on metric spaces
• Characterizations of Sobolev functions via Besov-type energy functionals in fractals
• Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces
• Schatten Properties of Calderón--Zygmund Singular Integral Commutator on stratified Lie groups
• Schatten--Lorentz characterization of Riesz transform commutator associated with Bessel operators
• BV functions and nonlocal functionals in metric measure spaces
• Sharp Besov capacity estimates for annuli in metric spaces with doubling measures
• Besov Spaces, Schatten Classes and Weighted Versions of the Quantised Derivative
• Solving a Dirichlet problem for unbounded domains via a conformal transformation
• Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces
• A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces
• Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups
• Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups
• Schatten classes and commutators of Riesz transform on Heisenberg group and applications
• New characterizations of Sobolev metric spaces
• Sobolev spaces and hyperbolic fillings
• The Poincaré inequality is an open ended condition
• Nearly weakly orthonormal sequences, singular value estimates, and Calderon-Zygmund operators

Cited by

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

Related

• Metric Sobolev spaces I: equivalence of definitions
• Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation
• Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted L^q spaces on metric spaces
• Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces
• p-Poincaré inequalities and cutoff Sobolev inequalities on metric measure spaces