TY - JOUR
AU - Alvarado, Ryan
AU - Hajłasz, Piotr
AU - Malý, Lukáš
PY - 2023/03/01
Y2 - 2023/05/31
TI - A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces
JF - Annales Fennici Mathematici
JA - Ann. Fenn. Math.
VL - 48
IS - 1
SE - Articles
DO - 10.54330/afm.127419
UR - https://afm.journal.fi/article/view/127419
SP - 255-275
AB - <pre>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 \(p\in (1,\infty)\).</pre><p> </p>
ER -