Geodesic
A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces
Ryan Alvarado1 ; Piotr Hajłasz2 ; Lukáš Malý3
1Amherst College, Department of Mathematics and Statistics
2University of Pittsburgh, Department of Mathematics
3Linköping University, Department of Science and Technology
Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 255-275

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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 $p\in (1,\infty)$.  
DOI : 10.54330/afm.127419
Keywords: Sobolev spaces, analysis on metric spaces, Poincaré inequality, uniform convexity

Ryan Alvarado  1   ; Piotr Hajłasz  2   ; Lukáš Malý  3

1 Amherst College, Department of Mathematics and Statistics
2 University of Pittsburgh, Department of Mathematics
3 Linköping University, Department of Science and Technology 
Ryan Alvarado; Piotr Hajłasz; Lukáš Malý. A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces. Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 255-275. doi: 10.54330/afm.127419
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