New Characterizations of First Order Sobolev Spaces

TL;DR

This paper introduces new characterizations of first order Sobolev spaces on metric measure spaces, establishing equivalences among various modified Sobolev spaces under mild conditions like Borel regularity and doubling measures.

Contribution

It provides novel characterizations of Sobolev spaces on metric measure spaces, showing equivalences among modified spaces under mild measure conditions.

Findings

Modified TC-Newtonian space is equivalent to Haj{ extl}asz-Sobolev space with Borel regular, sigma-finite measure.

All modified spaces are equivalent to Haj{ extl}asz-Sobolev space if the measure is doubling.

New characterizations hold under mild conditions, broadening the understanding of Sobolev spaces.

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: $\mathrm{TC}$-Newtonian space, $\hat{\mathrm{TC}}$-Newtonian space, and Gigli-like space. We prove that if the measure is Borel regular and $\sigma$-finite, then the modified $\mathrm{TC}$-Newtonian space is equivalent to the Haj{\l}asz-Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Haj{\l}asz-Sobolev space.