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Applications to Analysis on Quasimetric Spaces

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Book cover Groupoid Metrization Theory

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In this chapter we study the implications of our general metrization theory at the level of quasimetric spaces, with special emphasis on analytical aspects. More specifically, we study the nature of Hölder functions on quasimetric spaces by proving density, embeddings, separation, and extension theorems. We also quantify the richness of such spaces by introducing and studying a notion of index that interfaces tightly with the critical exponent beyond which the Hölder spaces become trivial. Other applications are targeted to Hardy spaces on spaces of homogeneous type, regularized distance, Whitney decompositions, and partitions of unity, as well as the Gromov–Pompeiu–Hausdorff distance.

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Notes

  1. 1.

    Trivially, if (X, d) is a metric space, then for each fixed x o  ∈ X the function \(d(\cdot,x_{o}) : X \rightarrow \mathbb{R}\) is Lipschitz.

  2. 2.

    What we here call the Pompeiu–Hausdorff distance has typically been referred to in the literature as the Hausdorff distance. For historical accuracy, however, it is significant to note that D. Pompeiu was the first to introduce (a slight version of) this concept in his thesis (written under the supervision of H. Poincaré). Pompeiu’s thesis appeared in print in [100], published in 1905, where Pompeiu calls this notion écart (mutuel) between two sets. Subsequently, in 1914, F. Hausdorff revisited this topic, and on p. 463 of his book [57] he correctly attributes the introduction of this notion to Pompeiu.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, USA

    Dorina Mitrea & Marius Mitrea

  2. Department of Mathematics, Temple University, Philadelphia, PA, USA

    Irina Mitrea

  3. UFR Sciences, Université Aix-Marseille III, Marseille, France

    Sylvie Monniaux

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Mitrea, D., Mitrea, I., Mitrea, M., Monniaux, S. (2013). Applications to Analysis on Quasimetric Spaces. In: Groupoid Metrization Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8397-9_4

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