Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices

TL;DR

This paper investigates Newtonian spaces based on quasi-Banach function lattices, proving the existence of minimal weak upper gradients and exploring their properties and convergence behaviors in abstract metric measure spaces.

Contribution

It establishes the existence of minimal weak upper gradients in Newtonian spaces based on quasi-Banach lattices and provides new representation formulas.

Findings

Existence of minimal weak upper gradients proven.

Representation formulas for weak upper gradients derived.

Convergence properties of Newtonian functions analyzed.

Abstract

Properties of first-order Sobolev-type spaces on abstract metric measure spaces, so-called Newtonian spaces, based on quasi-Banach function lattices are investigated. The set of all weak upper gradients of a Newtonian function is of particular interest. Existence of minimal weak upper gradients in this general setting is proven and corresponding representation formulae are given. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.