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Introduction

  • Dorina Mitrea
  • Irina Mitrea
  • Marius Mitrea
  • Sylvie Monniaux
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In this chapter we state a general metrization theorem, in the algebraic setting of groupoids, and explain its relationship to several classical results in analysis such as the Macías–Segovia metrization theorem for quasimetric spaces, the Aoki–Rolewicz theorem for quasinormed vector spaces, and the Alexandroff–Urysohn metrization theorem for uniform spaces. The metrization theorem in question is quantitative in nature and involves starting from a given quasisubadditive function defined on the underlying groupoid. We also indicate that our general metrization theorem is sharp.

Keywords

Hardy Space Topological Vector Space Uniform Space Homogeneous Type Normed Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This monograph has two distinct, yet closely interrelated, parts. In the first part (consisting of  Chaps. 1 – 3) we develop a metrization theory in the abstract setting of groupoids that, among other things, contains as particular cases the Aoki–Rolewicz theorem for locally bounded topological vector spaces and a sharpened version of the Macías–Segovia metrization theorem for quasimetric spaces. We also indicate how this theory can be used to provide a conceptually natural proof of the Alexandroff–Urysohn metrization theorem for uniform topological spaces. For this portion of our work, the methods employed are predominantly functional-analytic/algebraic, and the bulk of our results actually hold in the more general context of semigroupoids.

In the second part (comprised of  Chaps. 4– 6) we present a multitude of applications of our metrization theory to the area of analysis on quasimetric spaces (with a special emphasis on the structure and role of Hölder functions in such a setting), function space theory (covering topics such as completeness, embeddings, pointwise convergence, and separability of certain large classes of function spaces equipped with locally bounded, yet nonlocally convex, topologies), as well as classical functional analysis, dealing with open mapping and closed-graph-type theorems, and uniform boundedness principles, among other things, in settings where the notions of vector space and norm are significantly weakened. While precise details will be given later, here we wish to note that all our results hold in the class of quasi-Banach spaces. Regarding the significance of this category of spaces, in [67] N. Kalton writes that “There are sound reasons to want to develop understanding of these spaces, but the absence of one of the fundamental tools of functional analysis, the Hahn–Banach theorem, has proved a very significant stumbling block. However, there has been some progress in the non-convex theory and arguably it has contributed to our appreciation of Banach space theory.

1.1 Overview

In general, a topological space is said to be metrizable provided it is homeomorphic to a metric space. More transparently, a topological space (X, τ) is said to be metrizable if there exists a metric d on X with the property that the topology induced by d on X coincides with τ. As such, metrization theorems are results that give sufficient conditions for a topological space to be metrizable.

Consider three fundamental metrization results in various branches of mathematics:
  • The Alexandroff–Urysohn metrization theorem for uniform spaces (topology),

  • The Macías–Segovia metrization theorem for quasimetric spaces (harmonic analysis), and

  • The Aoki–Rolewicz theorem for quasinormed vector spaces (functional analysis).

The formal statement of the first theorem above is as follows (the reader is referred to future chapters for definitions clarifying the terminology employed here).

Theorem 1.1 (Alexandroff–Urysohn). 

Let X be a topological space. Then X is metrizable if and only if X is Hausdorff and the topology on X is induced by a uniform structure on X that has a countable fundamental system of entourages.

A related version of this theorem states that a uniform space is pseudometrizable (i.e., its topology is induced by a pseudometric) if and only if its uniformity has a countable base. See the discussion in J. Kelley’s book [71, Note 14, p. 186], where it is indicated that Theorem 1.1 originates in [4] (cf. also the discussion in Comment 2.82 in the last part of Sect. 2.2, which further underscores the prominent role of this classical result).

While the fact that the topology induced by a given quasidistance on a quasimetric space is metrizable is readily implied1 by Theorem 1.1 (something that was known long before Macías and Segovia’s work in [79]), Macías and Segovia’s main contribution was to bring to prominence the quantitative aspects of this result (in the setting of quasimetric spaces). This is apparent from an inspection of the statement of their theorem, which plays a basic role in the area of analysis on spaces of homogeneous type and which we recall below (as a slight reformulation of [79, Theorem 2, p. 259]).

Theorem 1.2 (Macías-Segovia). 

Let (X,ρ) be a quasimetric space, that is, X is a nonempty set and \(\rho: X \times X \rightarrow[0,+\infty )\) is a quasidistance, i.e., a function that, for every x,y,z ∈ X, satisfies 2
$$\rho (x,y) = 0\Longleftrightarrow\,x = y,\quad \rho (x,y) = \rho (y,x),\quad \rho (x,y) \leq c(\rho (x,z) + \rho (z,y)),$$
(1.1)
for some fixed finite constant c ≥ 1. Then there exists another quasidistance ρ on X that is equivalent to ρ (in the sense that each is dominated by a fixed multiple of the other) and satisfies the following additional properties. If
$$\begin{array}{rcl} \alpha:= \frac{1} {\mathrm{log}_{2}\,[c(2c + 1)]} \in(0,1),& &\end{array}$$
(1.2)
then the following assertions hold:
  1. (1)
    The function \({(\rho _{{\ast}})}^{\alpha } : X \times X \rightarrow[0,+\infty )\) satisfies
    $$\begin{array}{rcl} \rho _{{\ast}}{(x,y)}^{\alpha } \leq\rho _{ {\ast}}{(x,z)}^{\alpha } + \rho _{ {\ast}}{(z,y)}^{\alpha },\qquad \forall \,x,y,z \in X.& & \end{array}$$
    (1.3)
    Hence, \({(\rho _{{\ast}})}^{\alpha }\) is a distance on X that induces the same topology on X as the original quasidistance ρ. In particular, this topology is metrizable.
     
  2. (2)
    The function ρ satisfies the following Hölder-type regularity condition of order α:
    $$\begin{array}{c} \vert \rho _{{\ast}}(x,z) - \rho _{{\ast}}(y,z)\vert \leq\frac{1} {\alpha }\,\max \left \{\rho _{{\ast}}{(x,z)}^{1-\alpha },\rho _{ {\ast}}{(y,z)}^{1-\alpha }\right \}\rho _{ {\ast}}{(x,y)}^{\alpha }, \\ \forall \,x,y,z \in X\end{array}$$
    (1.4)
     

Ever since its original inception, Theorem 1.2 has played a pivotal role in analysis on spaces of homogeneous type since the natural setting for analysis in this context is that of quasimetric spaces. As noted earlier, the latter spaces are in fact metrizable, but it is a rather subtle matter to associate metrics, inducing the same topology, in a way that brings out the quantitative features of the quasimetric space in question in an optimal manner. The seminal work on this topic done by R. Macías and C. Segovia has been very influential in the study of spaces of homogeneous type. In particular, Theorem 1.2 is a popular result that has been widely cited; see, e.g., the discussion in the monographs [32] by Christ, [114] by Stein, [123] by Triebel, [59] by Heinonen, [56] by Han and Sawyer, [38] by David and Semmes, and [39] by Deng and Han, to name a few. Strictly speaking, Macías and Segovia’s original statement of this theorem has 3c 2 in place of c(2c + 1) in (1.2) but, as indicated in the discussion in Comment 2.83 at the end of Sect. 2.2, the number c(2c + 1) is the smallest constant for which their approach works as initially intended.

On to a different topic. It is well known that there are many function spaces of basic importance in partial differential equations that are not Banach but merely quasi-Banach. Indeed, this is the case for significant portions of the following familiar scales of spaces: Lebesgue spaces, weak Lebesgue spaces, Lorentz spaces, Hardy spaces, weak Hardy spaces, Lorentz-based Hardy spaces, Besov spaces, Triebel–Lizorkin spaces, and weighted versions of these spaces (among many others). In the context of quasinormed spaces, the Aoki–Rolewicz theorem reads as follows (the original references are [7, 103], and an excellent, timely exposition may be found in [69]).

Theorem 1.3 (Aoki–Rolewicz). 

Let X be a vector space equipped with a quasinorm \(\|\cdot \|\) , i.e., a nonnegative function defined on X that satisfies for each x,y ∈ X and each \(\lambda\in\mathbb{R}\)
$$\begin{array}{rcl} \|x\| = 0\Longleftrightarrow\,x = 0,\quad \|\lambda x\| = \vert \lambda \vert \|x\|,\quad \|x + y\| \leq c(\|x\| +\| y\|),& &\end{array}$$
(1.5)
for some fixed finite constant c ≥ 1. Then there exists a quasinorm \(\|\cdot \|_{{\ast}}\) on X that is equivalent to \(\|\cdot \|\) and is a p-norm for some p ∈ (0,1], i.e., it satisfies
$$\begin{array}{rcl} \|x + y\|_{{\ast}}^{p} \leq \| x\|_{ {\ast}}^{p} +\| y\|_{ {\ast}}^{p}\,\,\,\text{ for all }\,\,x,y \in X.& &\end{array}$$
(1.6)
In particular, the topology induced by \(\|\cdot \|\) on X is metrizable since it coincides with the topology induced by the (left- and right-invariant) distance \(d(x,y) :=\| x - y\|_{{\ast}}^{p}\) for all x,y ∈ X, on X.

Even though, strictly speaking, Theorems 1.2 and 1.3 are distinct results, it is inescapable that, at least formally, they share some basic characteristics (e.g., in [3, p. 319], the Macías–Segovia result is referred to as “an analogue” of the Aoki–Rolewicz theorem). This becomes even more apparent if Theorem 1.3 is reformulated in a (slightly more general, as it turns out) manner that places more emphasis on the quantitative aspects of the phenomenon at hand. For the reader’s convenience, we will first momentarily digress for the purpose of recalling some basic facts and terminology from the theory of topological vector spaces.

A topological vector space is said to be locally bounded provided there exists a (topologically) bounded neighborhood of the zero vector. Recall that, in this context, being bounded means that the set in question is absorbed by each neighborhood of zero (and not that it is geometrically bounded, in the sense of having a finite diameter). Specifically, E is (topologically) bounded if and only if for every neighborhood V of the zero vector there exists a real number \(\lambda _{{\ast}} > 0\) such that \(E \subseteq \lambda V\) for every scalar \(\lambda> \lambda _{{\ast}}\). Also (cf., e.g., [74, (1) p. 159]), a topological vector space X is locally bounded if and only if there exists a quasinorm \(\|\cdot \|\) on X that yields the same topology on X as the original one (which amounts to the condition that the balls \(\{y \in X :\,\| y\| < r\}\), \(r \in(0,+\infty )\), constitute a fundamental system of neighborhoods for the zero vector). Moreover, a set in a quasinormed space is bounded in the topology induced by the quasinorm if and only if it has finite diameter with respect to the quasinorm. Hence, a quasi-Banach space is a complete, locally bounded topological vector space. Let us also recall that a set E in a vector space is said to be balanced if \(\lambda E \subseteq E\) for every scalar λ with \(\vert \lambda \vert \leq1\). Remarkably, a quasinormed vector space is locally convex if and only if it is linearly isomorphic to a normed vector space (local convexity signifies the existence of a fundamental system of neighborhoods for the zero vector consisting of absorbing, balanced, and convex sets; one convenient description of the fact that a subset E of a vector space is convex is that \((\lambda+ \eta )E = \lambda E + \eta E\) for all scalars λ, η > 0). Compare, for example, [69] for more details.

Returning to the mainstream discussion, we record the following more precise version of the Aoki–Rolewicz theorem (again, see the informative discussion in [69]).

Theorem 1.4 (Aoki–Rolewicz). 

Let X be a Hausdorff, locally bounded, topological vector space. In particular, there exists a bounded and balanced neighborhood B of the zero vector in X. Let \(c \in(1,+\infty )\) be such that \(B + B \subseteq cB\) , and define
$$\begin{array}{rcl} p := \frac{1} {\log _{2}c} \in(0,+\infty ).& &\end{array}$$
(1.7)
Finally, for each x ∈ X, set
$$\begin{array}{rcl} \|x\| :=\inf \left \{{\left (\sum\limits_{i=1}^{N}\vert \!\vert \!\vert x_{ i}\vert \!\vert \!\vert _{B}^{p}\right )}^{\frac{1} {p} } :\, N \in\mathbb{N},\,\,x_{1},\ldots ,x_{N} \in X\,\,\text{ such that}\,\,\sum\limits_{i=1}^{N}x_{ i} = x\right \},& & \\ & &\end{array}$$
(1.8)
where \(\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert _{B}\) is the Minkowski gauge function associated with B, i.e.,
$$\begin{array}{rcl} \vert \!\vert \!\vert x\vert \!\vert \!\vert _{B} :=\inf \,\{ \lambda> 0 :\, {\lambda }^{-1}x \in B\},\qquad \forall \,x \in X.& &\end{array}$$
(1.9)

Then \(\|\cdot \|\) defined in (1.8) is a p-norm on the vector space X, which is equivalent to the quasinorm \(\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert _{B}\) (and, hence, induces the same topology on X as the original one). As a consequence, X is a locally p-convex vector space whenever c > 2 and a locally convex vector space whenever \(c \leq2\) , and the topology on X is metrizable via a two-sided invariant distance.

On the face of the evidence presented so far, an optimistic observer would hope that the formal analogies between the statements of Theorems 1.2 and 1.4 would indicate that there is a more general phenomenon at work here encompassing the named results as particular manifestations. In this vein, it is worth recalling a popular dictum of E.H. Moore to the effect that whenever there are parallel theories, typically there is one that subsumes them all.

One of the goals of the present monograph is to shed light on this issue by proving a metrization theorem that contains both Theorem 1.2 and Theorem 1.4 (hence also Theorem 1.3) in a canonical fashion and that may also be used to provide a conceptually natural proof of Theorem 1.1. We manage to accomplish this without compromising the sharpness of the quantitative aspects of the results in question (for example, even when specialized to the particular case of quasimetric spaces our results yield a significant improvement of Theorem 1.2) and, also, are able to work under minimal algebraic assumptions, which ensures a desirable degree of versatility for our result. The latter aspect is particularly important for applications, as will become apparent from the discussion in  Chaps. 4– 6, where the impact of this metrization theory on other branches of mathematics is brought to light.

The unifying language that permits such a generalization is that of groupoids. Recall that the concept of groupoid was originally introduced by H. Brandt3 in 1926 as an algebraic structure generalizing the notion of group by allowing the multiplication to be just partially defined (for more on this topic see the discussion in Sect. 2).

1.2 First Look at the Groupoid Metrization Theorem

A sample of the metrization results proved here in the context of groupoids is as follows (the body of the monograph contains stronger results in the sense that they indicate what can be achieved with weaker, or fewer, assumptions; see also Theorem 3.26 for a substantially expanded version of this result).

Theorem 1.5.

Let \((G,{\ast},{(\cdot )}^{-1})\) be a groupoid, with partial multiplication ∗ and inverse operation \({(\cdot )}^{-1}\) . For each \(N \in\mathbb{N}\) denote by G (N) the set of all ordered N-tuples of elements in G whose product (in the given order) is meaningfully defined. Furthermore, denote by G (0) the unit space of G, and introduce \({\mathcal{G}}^{\mathrm{R}} :=\{ (a,b) \in G \times G :\, (a,{b}^{-1}) \in{G}^{(2)}\}\) .

Next, assume that \(\psi: G \rightarrow[0,+\infty )\) is a function for which there exist two finite constants \(C_{0} \geq0\) and \(C_{1} \geq1\) such that the following properties hold:
$$\begin{array}{rcl} & & \bullet \,\text{ quasisubadditivity:}\quad \psi (a {\ast} b) \leq C_{1}\max \{\psi (a),\psi (b)\},\quad \text{ for all }\,\,(a,b) \in{G}^{(2)},\end{array}$$
(1.10)
$$\begin{array}{rcl} & & \bullet \,\text{ quasisymmetry:}\quad \psi ({a}^{-1}) \leq C_{ 0}\,\psi (a),\qquad \text{ for every }\,\,a \in G,\end{array}$$
(1.11)
$$\begin{array}{rcl} & & \bullet \,\text{ nondegeneracy:}\quad a \in G\text{ and }\psi (a) = 0\, \Leftrightarrow \, a \in{G}^{(0)},\text{ i.e., }{\psi }^{-1}(\{0\}) = {G}^{(0)}.\end{array}$$
(1.12)
Denote by \(\tau _{\psi }^{\mathrm{R}}\) the right topology induced by ψ on G, defined as the largest topology on G with the property that for any element a ∈ G a fundamental system of neighborhoods is given by \(\{B_{\psi }^{\mathrm{R}}(a,r)\}_{r>0}\) , where for each \(r \in(0,+\infty )\),
$$\begin{array}{rcl} B_{\psi }^{\mathrm{R}}(a,r) := \left \{b \in G :\, (a,b) \in {\mathcal{G}}^{\mathrm{R}}\text{ and }\psi (a \ast {b}^{-1}) < r\right \}.& &\end{array}$$
(1.13)
Also, with C 1 ≥ 1 as in (1.10), let
$$\begin{array}{rcl} \alpha:= \frac{1} {\log _{2}C_{1}} \in(0,+\infty ].& &\end{array}$$
(1.14)
Finally, introduce a symmetrized version of ψ by setting
$$\begin{array}{rcl} \psi _{\mathrm{sym}}(a) :=\max \{ \psi (a),\psi ({a}^{-1})\},\quad \forall \,a \in G,& &\end{array}$$
(1.15)
and define the canonical regularization \(\psi _{\mathrm{reg}} : G \rightarrow[0,+\infty )\) of ψ by considering, for each a ∈ G,
$$\begin{array}{rcl} \psi _{\mathrm{reg}}(a)& :=& \inf \,\left \{{\left (\sum\limits_{i=1}^{N}\psi _{\mathrm{ sym}}{(a_{i})}^{\alpha }\right )}^{ \frac{1} {\alpha } } :\, N \in\mathbb{N},\right . \\ & & \qquad \quad \left .(a_{1},\ldots ,a_{N}) \in{G}^{(N)},\,\,a = a_{ 1} {\ast}\cdots{\ast} a_{N}\right \}\end{array}$$
(1.16)
(with a natural alteration in the case when \(\alpha= +\infty \) ).
Then the following conclusions hold.
  1. (1)
    The function ψ reg is symmetric, in the sense that
    $$\begin{array}{rcl} \psi _{\mathrm{reg}}({a}^{-1}) = \psi _{\mathrm{ reg}}(a)\,\,\,\text{ for every }\,\,a \in G,& & \end{array}$$
    (1.17)
    and ψ reg is quasisubadditive, in the precise sense that, with C 1 denoting the same constant as in (1.10), one has
    $$\psi _{\mathrm{reg}}(a {\ast} b) \leq C_{1}\max \{\psi _{\mathrm{reg}}(a),\psi _{\mathrm{reg}}(b)\}\quad \text{ for all }\,\,(a,b) \in{G}^{(2)}.$$
    (1.18)
     
  2. (2)
    With C 0 and C 1 as in (1.10) and (1.11), there holds
    $$\begin{array}{rcl} C_{1}^{-2}\psi\leq\psi _{\mathrm{ reg}} \leq \max \,\{ 1,C_{0}\}\,\psi \quad \text{ on }\,\,G.& & \end{array}$$
    (1.19)
    In particular, \(\psi _{\mathrm{reg}}^{-1}(\{0\}) = {G}^{(0)}\).
     
  3. (3)
    For each β ∈ (0,α] the function ψ reg is β-subadditive in the sense that one has (with a natural interpretation when \(\beta= \alpha= +\infty \) )
    $$\begin{array}{rcl} \psi _{\mathrm{reg}}(a {\ast} b) \leq {\left (\psi _{\mathrm{reg}}{(a)}^{\beta } + \psi _{\mathrm{ reg}}{(b)}^{\beta }\right )}^{\frac{1} {\beta } },\qquad \forall \,(a,b) \in{G}^{(2)}.& & \end{array}$$
    (1.20)
     
  4. (4)
    For each finite number β ∈ (0,α] the function ψ reg satisfies the following Hölder-type regularity condition of order β:
    $$\begin{array}{rcl} \left \vert \psi _{\mathrm{reg}}(a) - \psi _{\mathrm{reg}}(b)\right \vert \leq\frac{1} {\beta }\,\max \,\left \{\psi _{\mathrm{reg}}{(a)}^{1-\beta },\psi _{\mathrm{ reg}}{(b)}^{1-\beta }\right \}{\left [\psi _{\mathrm{ reg}}(a \ast {b}^{-1})\right ]}^{\beta }\qquad & & \end{array}$$
    (1.21)
    whenever \((a,b) \in {\mathcal{G}}^{\mathrm{R}}\) (with the understanding that when β ≥ 1, one also imposes the condition that \(a,b\not\in {G}^{(0)}\)). Furthermore, the upper bound (1.14) for the exponent β appearing in this Hölder-type regularity result is sharp.
     
  5. (5)

    The function \(\psi _{\mathrm{reg}} : \left (G,\tau _{\psi }^{\mathrm{R}}\right ) \rightarrow[0,+\infty )\) is continuous, and for every a ∈ G and r > 0 the right ψ reg -ball \(B_{\psi _{\mathrm{reg}}}^{\mathrm{R}}(a,r) :=\{ b \in G :\, (a,b) \in {\mathcal{G}}^{\mathrm{R}}\text{ and }\psi _{\mathrm{ reg}}(a \ast {b}^{-1}) < r\}\) is open in the topology \(\tau _{\psi }^{\mathrm{R}}\).

     
  6. (6)
    For each finite number β ∈ (0,α] define the function
    $$\begin{array}{rcl} d_{\psi ,\beta }^{\mathrm{R}} : {\mathcal{G}}^{\mathrm{R}} \rightarrow[0,+\infty ),\,\,\quad d_{ \psi ,\beta }^{\mathrm{R}}(a,b) :={ \left [\psi _{\mathrm{ reg}}(a \ast {b}^{-1})\right ]}^{\beta },\quad \forall \,(a,b) \in {\mathcal{G}}^{\mathrm{R}}.& & \end{array}$$
    (1.22)
    Then d ψ,β R is a partially defined distance on G with domain \({\mathcal{G}}^{\mathrm{R}}\) , i.e., it satisfies the following conditions:
    $$\begin{array}{rcl} & & \mathit{for\ any}\ (a,b) \in {\mathcal{G}}^{\mathrm{R}},\ \mathit{one\ has}\ d_{ \psi ,\beta }^{\mathrm{R}}(a,b) = 0\ \mathit{if \ and\ only\ if }\ a = b, \\ & & d_{\psi ,\beta }^{\mathrm{R}}(a,b) = d_{ \psi ,\beta }^{\mathrm{R}}(b,a)\ \mathit{for\ every}\ (a,b) \in {\mathcal{G}}^{\mathrm{R}}, \\ & & d_{\psi ,\beta }^{\mathrm{R}}(a,b) \leq d_{ \psi ,\beta }^{\mathrm{R}}(a,c) + d_{ \psi ,\beta }^{\mathrm{R}}(c,b)\ \mathit{for\ all}\ (a,c),(c,b) \in {\mathcal{G}}^{\mathrm{R}}. \end{array}$$
    (1.23)
    Moreover, the topology induced by the partially defined distance \(d_{\psi ,\beta }^{\mathrm{R}}\) on G is  \(\tau _{\psi }^{\mathrm{R}}\).
     
  7. (7)
    The partially defined distance \(d_{\psi ,\beta }^{\mathrm{R}}\) introduced in (1.22) is right-invariant, in the sense that
    $$\begin{array}{c} (a,b) \in {\mathcal{G}}^{\mathrm{R}}\text{ and}\ c \in G\text{ such that }(a,c),(b,c) \in{G}^{(2)} \\ \Longrightarrow\,(a {\ast} c,b {\ast} c) \in {\mathcal{G}}^{\mathrm{R}}\text{ and }d_{ \psi ,\beta }^{\mathrm{R}}(a {\ast} c,b {\ast} c) = d_{ \psi ,\beta }^{\mathrm{R}}(a,b)\end{array}$$
    (1.24)
    In the particular case when G is a group, \({\mathcal{G}}^{\mathrm{R}} = G \times G\) and, hence, the function \(d_{\psi ,\beta }^{\mathrm{R}}\) is a genuine right-invariant distance on G.
     

Theorem 1.5 contains Macías and Segovia’s metrization result formulated in Theorem 1.2 when specialized to the particular case when the groupoid G is the so-called pair groupoid X ×X associated with the ambient set X (as described in Example 2.31) of a quasimetric space. Moreover, Theorem 1.5 subsumes Aoki and Rolewicz’s metrization result stated in Theorem 1.4 in the scenario in which the groupoid G is the underlying (Abelian) additive group of a given vector space X (cf. Example 2.29). The interplay between these results is studied in more detail in the body of the monograph; see the discussion in Sect. 3.2.3 in this regard. In particular, here we also elaborate on the manner in which Theorem 1.5 contains the Alexandroff–Urysohn metrization theorem (formulated in Theorem 1.1).

We wish to stress that the actual optimal value of the Hölder regularity exponent α (playing the role of upper bound of βs for which (1.21) holds) is not an issue of mere curiosity since this number plays a most fundamental role in the theory of function spaces that can be developed on spaces of homogeneous type. For example, the issue of identifying the sharp value of the Hölder regularity exponent α from (1.4) is raised explicitly in Remark 5.3 on p. 133 of [62], where the reader may find more details pertaining to the case of Hardy spaces. Here we wish to note that, when combined with the work in [80], our results lead to a satisfactory theory for Hardy spaces H p (X) whenever the quasimetric space (X, ρ) is equipped with an Ahlfors–David regular measure μ of order d > 0 and
$$\begin{array}{rcl} \frac{d} {d +\min \,\{ d,{[\mathrm{log}_{2}C_{\rho }]}^{-1}\}} < p \leq1,& &\end{array}$$
(1.25)
where C ρ is the optimal constant in the inequality \(\rho (x,y) \leq C\,\max \,\{\rho (x,z),\rho (z,y)\}\) for all x, y, z ∈ X (see Theorem 4.102 for details). It is worth remarking that this range for p is in the nature of best possible since from (1.25) we recover the familiar condition \(\frac{n} {n+1} < p \leq1\) (associated with atomic Hardy spaces for atoms satisfying one vanishing moment condition) in the case when \(X := {\mathbb{R}}^{n}\), \(n \in\mathbb{N}\), equipped with the Euclidean distance and the n-dimensional Lebesgue measure. In fact, similar considerations apply to the case of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, as discussed in [39, 56]and others.

Another perspective that highlights the usefulness of a sharp Hölder regularity exponent α (in the context of (1.4)) is as follows. On the one hand, one naturally expects to have α = 1 in the case when (X, ρ) is actually a metric space, since a distance function is Lipschitz in each of its variables. On the other hand, in the setting of Theorem 1.2, the condition that ensures that (X, ρ) is a metric space is c = 1, and, according to (1.2)–(1.4), this only yields the generally unsatisfactory result that a distance function is Hölder continuous of order 1 ∕ log2 3. By way of contrast, the value of α in (1.14) becomes, as expected, 1 when C 1 = 2.

Our approach builds on and extends the work of Peetre and Sparr [97] (in the setting of normed Abelian groups), Gustavsson [53] (where a metrization theorem for semigroupoids is proved for a nonoptimal exponent α, namely \(\alpha= {(2\,\mathrm{log}_{2}\,C_{1})}^{-1}\), i.e., half the value of α in (3.190)), and the classical work of Frink [49]. For a more in-depth discussion elaborating on the connections between Theorems 1.5 and 1.11.4, which also provides further motivational examples and background, the reader is referred to Sects. 3.2.3 and 3.2.4.

The organization of the monograph is as follows. The material in Sects. 2.1.1 and 2.1.2 amounts to a concise (yet self-contained) introduction to the theory of semigroupoids and groupoids, and in Sect. 2.2 we review topics of a topological flavor. The bulk of the work pertaining to quantitative metrization results is concentrated in  Chap. 3. In particular, the regularization results for quasisubadditive mappings established in Sect. 3.1 greatly facilitate the presentation of our main groupoid metrization theorem. The latter is stated in Sect. 3.2.1 and proved in Sect. 3.2.2, and its various connections with Macías–Segovia, Aoki–Rolewicz, and Alexandroff–Urysohn theorems are highlighted in Sect. 3.2.3. The scope of this result is further expanded in Sect. 3.3.1 to the setting of semigroupoids. Several applications of this semigroupoid metrization theory are subsequently discussed in Sects. 3.3.2 and 3.3.3. Next, in Sect. 3.4, we state and prove a sharpened version of the Macías–Segovia result; cf. Theorem 3.46.

Moving on, in  Chap. 4, we present a significant number of applications of our metrization theorems to analysis on quasimetric spaces. Without going into detail, the list of topics considered in this chapter includes extensions of Hölder functions, separation, density and embedding properties of Hölder functions, the regularized distance function to a set, Whitney-like partitions of unity via Hölder functions, the smoothness indexes of a quasimetric space, distribution theory on quasimetric spaces, Hardy spaces on Ahlfors-regular quasimetric spaces, approximation to the identity on Ahlfors-regular quasimetric spaces, bi-Lipschitz Euclidean embeddings of quasimetric spaces, the quasimetric version of Kuratowski’s and Fréchet’s embedding theorems, the Pompeiu–Hausdorff quasidistance on quasimetric spaces, and the Gromov–Pompeiu–Hausdorff distance between quasimetric spaces.

 Chapter 5 is devoted to presenting applications of the metrization theory developed in  Chap. 3 to function space theory, with a special emphasis on topics such as completeness, embeddings, pointwise convergence, and separability of certain inclusive classes of function spaces endowed with locally bounded, yet nonlocally convex, topologies. Finally, in  Chap. 6 we revisit some of the cornerstones of classical functional analysis (including open mapping and closed-graph-type theorems, as well as uniform boundedness principles) in settings where the traditional context of a normed vector space is significantly relaxed. Once again, our metrization theory developed in the earlier chapters plays a key role in this endeavor.

Footnotes

  1. 1.

    Any quasimetric space (X, ρ) may be canonically viewed as a uniform space whose uniformity has a countable fundamental system of entourages, say, \(\left \{(x,y) \in X \times X :\, \rho (x,y) < {n}^{-1}\right \}\), \(n \in\mathbb{N}\).

  2. 2.

    The interested reader is referred to [99] for historical references pertaining to quasinormed spaces.

  3. 3.

    Strictly speaking, in [21] Brandt introduced a smaller class of groupoids, i.e., what is nowadays referred to as transitive groupoids.

Notes

Acknowledgements

In the early stages of its inception, various parts of this monograph were used to teach several topic courses at the graduate level at the University of Missouri. We wish to take this opportunity to thank our students, especially Ryan Alvarado, Kevin Brewster, Dan Brigham, Brock Schmutzler, and Elia Ziadé, for their active participation and their careful reading of preliminary notes. The authors also gratefully acknowledge the support of the Simons Foundation Grant No. 200750 as well as the US NSF Grants DMS-1201736 and DMS-0653180. Last but not least, the authors thank the anonymous referees for making a number of useful suggestions, which have improved the presentation.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Dorina Mitrea
    • 1
  • Irina Mitrea
    • 2
  • Marius Mitrea
    • 1
  • Sylvie Monniaux
    • 3
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA
  3. 3.UFR SciencesUniversité Aix-Marseille IIIMarseilleFrance

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