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

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.

References in corpus (1)

• Newtonian spaces based on quasi-Banach function lattices

Cited by in corpus (6)

• On the properties of quasi-Banach function spaces
• An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces
• On the properties of rearrangement-invariant quasi-Banach function spaces
• Hajlasz Gradients Are Upper Gradients
• Newtonian spaces based on quasi-Banach function lattices
• Morrey-Sobolev Spaces on Metric Measure Spaces