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Lipschitz-free Spaces on Finite Metric Spaces

  • Stephen J. Dilworth (a1), Denka Kutzarova (a2) (a3) and Mikhail I. Ostrovskii (a4)

Abstract

Main results of the paper are as follows:

(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$ .

(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.

Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

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Copyright

Footnotes

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Author S. D. was supported by the National Science Foundation under Grant Number DMS–1361461. Authors S. D. and D. K. were supported by the Workshop in Analysis and Probability at Texas A&M University in 2017. Author M. O. was supported by the National Science Foundation under Grant Number DMS–1700176.

Footnotes

References

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Lipschitz-free Spaces on Finite Metric Spaces

  • Stephen J. Dilworth (a1), Denka Kutzarova (a2) (a3) and Mikhail I. Ostrovskii (a4)

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