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

Ryan Alvarado¹ ; Piotr Hajłasz² ; Lukáš Malý³

¹ Amherst College, Department of Mathematics and Statistics
² University of Pittsburgh, Department of Mathematics
³ Linköping University, Department of Science and Technology

Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 255-275.

Résumé

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 $N^{1,p}(X)$ Sobolev space is reflexive and separable whenever $p\in (1,\infty)$. We also prove separability of the space when $p=1$. Our proof is based on a straightforward construction of an equivalent norm on $N^{1,p}(X)$, $p\in [1,\infty)$, that is uniformly convex when $p\in (1,\infty)$. Finally, we explicitly construct a functional that is pointwise comparable to the minimal $p$-weak upper gradient, when $p\in (1,\infty)$.  

Détail

DOI : 10.54330/afm.127419

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