Isometric classification of Sobolev spaces on graphs

Tom 109 / 2007

M. I. Ostrovskii Colloquium Mathematicum 109 (2007), 287-295 MSC: 52A21, 46B04, 05C40. DOI: 10.4064/cm109-2-10


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 $\mathcal{G}\times\mathbb{Z}_2$.


  • M. I. OstrovskiiDepartment of Mathematics and Computer Science
    St. John's University
    8000 Utopia Parkway
    Queens, NY 11439, U.S.A.

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