Newtonian spaces based on quasi-Banach function lattices

TL;DR

This paper introduces Newtonian spaces on metric measure spaces using quasi-Banach function lattices to generalize Sobolev spaces, developing tools like moduli and capacities to analyze their properties.

Contribution

It defines Newtonian spaces based on quasi-Banach lattices and establishes fundamental properties such as absolute continuity and completeness.

Findings

Newtonian spaces are well-defined on metric measure spaces.

Basic properties like absolute continuity along curves are proven.

Completeness of these spaces is established.

Abstract

In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.