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Embeddability of Snowflaked Metrics with Applications to the Nonlinear Geometry of the Spaces L p and ℓ p for 0<p<∞

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Abstract

We study the classical spaces L p and ℓ p for the whole range 0<p<∞ from a metric viewpoint. As we go along, we look over some of the results and techniques that, together with our work in this paper, have permitted us to obtain a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with their ad hoc distances and their snowflakings. Through connections with weaker forms of embeddings that lead to basic (yet fundamental) open problems, we also set the challenging goal of understanding the dissimilarities between the well-known subspace structure and the different nonlinear geometries that coexist inside L p and ℓ p .

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Acknowledgements

The second author is very grateful to Bill Johnson and Gideon Schechtman for various enlightening discussions on the topics covered in this article.

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Authors and Affiliations

  1. Mathematics Department, Universidad Pública de Navarra, Pamplona, 31006, Spain

    F. Albiac

  2. Mathematics Department, Texas A&M University, College Station, TX, 77843, USA

    F. Baudier

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  1. F. Albiac
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  2. F. Baudier
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Correspondence to F. Albiac.

Additional information

Communicated by Loukas Grafakos.

This article grew out of the authors’ stay at the Mathematical Sciences Research Institute in Berkeley, California, as research members in the Quantitative Geometry program held in the fall of 2011, partially supported by NSF grant DMS-0932078 administered by the MSRI. The authors also acknowledge the support of the Spanish Ministerio de Ciencia e Innovación Research Grant Operadores, retículos, y geometría de espacios de Banach, reference number MTM2008-02652/MTM.

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Albiac, F., Baudier, F. Embeddability of Snowflaked Metrics with Applications to the Nonlinear Geometry of the Spaces L p and ℓ p for 0<p<∞. J Geom Anal 25, 1–24 (2015). https://doi.org/10.1007/s12220-013-9390-0

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