New characterizations of first order Sobolev spaces
Abstract
Abstract We provide new characterizations of Sobolev spaces that are true under some mild conditions. We study modified first order Sobolev spaces on metric measure spaces: $${{\,\mathrm{\textrm{TC}}\,}}$$ TC -Newtonian space, $$\hat{{{\,\mathrm{\textrm{TC}}\,}}}$$ TC ^ -Newtonian space, and Ambrosio–Gigli–Savaré-like space. We prove that if the measure is Borel regular and $$\sigma $$ σ -finite, then the modified $${{\,\mathrm{\textrm{TC}}\,}}$$ TC -Newtonian space is equivalent to the Hajłasz–Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Hajłasz–Sobolev space.
Fields of Study & Subjects
Advanced Harmonic Analysis Research
Differential Equations and Boundary Problems
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- Annals of Functional Analysis
- Volume
- 17
- Issue
- 3
- Publication date
- 19 June 2026
- Language
- EN