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

Kirjoittajat

  • 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

Avainsanat:

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

Abstrakti

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 \(p\in (1,\infty)\).

 

Osasto
Articles

Julkaistu

2023-03-01

Viittaaminen

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