Isometric classification of Sobolev spaces on graphs
Tom 109 / 2007
Colloquium Mathematicum 109 (2007), 287-295
MSC: 52A21, 46B04, 05C40.
DOI: 10.4064/cm109-2-10
Streszczenie
Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach–Stone theorem is valid: if two Sobolev spaces on $3$-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group $\mathcal{G}$ and each $p$ which is not an even integer, there exists $n\in\mathbb{N}$ and a subspace $L\subset\ell_p^n$ whose group of isometries is the direct product $\mathcal{G}\times\mathbb{Z}_2$.
