A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces

Authors

  • Ryan Alvarado Amherst College, Department of Mathematics and Statistics
  • Piotr Hajłasz University of Pittsburgh, Department of Mathematics
  • Lukáš Malý Linköping University, Department of Science and Technology

Keywords:

Sobolev spaces, analysis on metric spaces, Poincaré inequality, uniform convexity

Abstract

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space X supports a p-Poincaré inequality, then the N1,p(X) Sobolev space is reflexive and separable whenever p(1,). We also prove separability of the space when p=1. Our proof is based on a straightforward construction of an equivalent norm on N1,p(X), p[1,), that is uniformly convex when p(1,). Finally, we explicitly construct a functional that is pointwise comparable to the minimal p-weak upper gradient, when p(1,).

 

Section
Articles

Published

2023-03-01

How to Cite

Alvarado, R., Hajłasz, P., & Malý, L. (2023). A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces. Annales Fennici Mathematici, 48(1), 255–275. https://doi.org/10.54330/afm.127419