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


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.


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
$$\rho (x,y) = 0\Longleftrightarrow\,x = y,\quad \rho (x,y) = \rho (y,x),\quad \rho (x,y) \leq c(\rho (x,z) + \rho (z,y)),$$
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
$$\begin{array}{rcl} \alpha:= \frac{1} {\mathrm{log}_{2}\,[c(2c + 1)]} \in(0,1),& &\end{array}$$
then the following assertions hold:
  1. (1)
    The function \({(\rho _{{\ast}})}^{\alpha } : X \times X \rightarrow[0,+\infty )\) satisfies
    $$\begin{array}{rcl} \rho _{{\ast}}{(x,y)}^{\alpha } \leq\rho _{ {\ast}}{(x,z)}^{\alpha } + \rho _{ {\ast}}{(z,y)}^{\alpha },\qquad \forall \,x,y,z \in X.& & \end{array}$$
    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:
    $$\begin{array}{c} \vert \rho _{{\ast}}(x,z) - \rho _{{\ast}}(y,z)\vert \leq\frac{1} {\alpha }\,\max \left \{\rho _{{\ast}}{(x,z)}^{1-\alpha },\rho _{ {\ast}}{(y,z)}^{1-\alpha }\right \}\rho _{ {\ast}}{(x,y)}^{\alpha }, \\ \forall \,x,y,z \in X\end{array}$$

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 \(\lambda\in\mathbb{R}\)
$$\begin{array}{rcl} \|x\| = 0\Longleftrightarrow\,x = 0,\quad \|\lambda x\| = \vert \lambda \vert \|x\|,\quad \|x + y\| \leq c(\|x\| +\| y\|),& &\end{array}$$
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
$$\begin{array}{rcl} \|x + y\|_{{\ast}}^{p} \leq \| x\|_{ {\ast}}^{p} +\| y\|_{ {\ast}}^{p}\,\,\,\text{ for all }\,\,x,y \in X.& &\end{array}$$
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
$$\begin{array}{rcl} p := \frac{1} {\log _{2}c} \in(0,+\infty ).& &\end{array}$$
Finally, for each x ∈ X, set
$$\begin{array}{rcl} \|x\| :=\inf \left \{{\left (\sum\limits_{i=1}^{N}\vert \!\vert \!\vert x_{ i}\vert \!\vert \!\vert _{B}^{p}\right )}^{\frac{1} {p} } :\, N \in\mathbb{N},\,\,x_{1},\ldots ,x_{N} \in X\,\,\text{ such that}\,\,\sum\limits_{i=1}^{N}x_{ i} = x\right \},& & \\ & &\end{array}$$
where \(\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert _{B}\) is the Minkowski gauge function associated with B, i.e.,
$$\begin{array}{rcl} \vert \!\vert \!\vert x\vert \!\vert \!\vert _{B} :=\inf \,\{ \lambda> 0 :\, {\lambda }^{-1}x \in B\},\qquad \forall \,x \in X.& &\end{array}$$

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:
$$\begin{array}{rcl} & & \bullet \,\text{ quasisubadditivity:}\quad \psi (a {\ast} b) \leq C_{1}\max \{\psi (a),\psi (b)\},\quad \text{ for all }\,\,(a,b) \in{G}^{(2)},\end{array}$$
$$\begin{array}{rcl} & & \bullet \,\text{ quasisymmetry:}\quad \psi ({a}^{-1}) \leq C_{ 0}\,\psi (a),\qquad \text{ for every }\,\,a \in G,\end{array}$$
$$\begin{array}{rcl} & & \bullet \,\text{ nondegeneracy:}\quad a \in G\text{ and }\psi (a) = 0\, \Leftrightarrow \, a \in{G}^{(0)},\text{ i.e., }{\psi }^{-1}(\{0\}) = {G}^{(0)}.\end{array}$$
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 \(r \in(0,+\infty )\),
$$\begin{array}{rcl} B_{\psi }^{\mathrm{R}}(a,r) := \left \{b \in G :\, (a,b) \in {\mathcal{G}}^{\mathrm{R}}\text{ and }\psi (a \ast {b}^{-1}) < r\right \}.& &\end{array}$$
Also, with C 1 ≥ 1 as in (1.10), let
$$\begin{array}{rcl} \alpha:= \frac{1} {\log _{2}C_{1}} \in(0,+\infty ].& &\end{array}$$
Finally, introduce a symmetrized version of ψ by setting
$$\begin{array}{rcl} \psi _{\mathrm{sym}}(a) :=\max \{ \psi (a),\psi ({a}^{-1})\},\quad \forall \,a \in G,& &\end{array}$$
and define the canonical regularization \(\psi _{\mathrm{reg}} : G \rightarrow[0,+\infty )\) of ψ by considering, for each a ∈ G,
$$\begin{array}{rcl} \psi _{\mathrm{reg}}(a)& :=& \inf \,\left \{{\left (\sum\limits_{i=1}^{N}\psi _{\mathrm{ sym}}{(a_{i})}^{\alpha }\right )}^{ \frac{1} {\alpha } } :\, N \in\mathbb{N},\right . \\ & & \qquad \quad \left .(a_{1},\ldots ,a_{N}) \in{G}^{(N)},\,\,a = a_{ 1} {\ast}\cdots{\ast} a_{N}\right \}\end{array}$$
(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
    $$\begin{array}{rcl} \psi _{\mathrm{reg}}({a}^{-1}) = \psi _{\mathrm{ reg}}(a)\,\,\,\text{ for every }\,\,a \in G,& & \end{array}$$
    and ψ reg is quasisubadditive, in the precise sense that, with C 1 denoting the same constant as in (1.10), one has
    $$\psi _{\mathrm{reg}}(a {\ast} b) \leq C_{1}\max \{\psi _{\mathrm{reg}}(a),\psi _{\mathrm{reg}}(b)\}\quad \text{ for all }\,\,(a,b) \in{G}^{(2)}.$$
  2. (2)
    With C 0 and C 1 as in (1.10) and (1.11), there holds
    $$\begin{array}{rcl} C_{1}^{-2}\psi\leq\psi _{\mathrm{ reg}} \leq \max \,\{ 1,C_{0}\}\,\psi \quad \text{ on }\,\,G.& & \end{array}$$
    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 \(\beta= \alpha= +\infty \) )
    $$\begin{array}{rcl} \psi _{\mathrm{reg}}(a {\ast} b) \leq {\left (\psi _{\mathrm{reg}}{(a)}^{\beta } + \psi _{\mathrm{ reg}}{(b)}^{\beta }\right )}^{\frac{1} {\beta } },\qquad \forall \,(a,b) \in{G}^{(2)}.& & \end{array}$$
  4. (4)
    For each finite number β ∈ (0,α] the function ψ reg satisfies the following Hölder-type regularity condition of order β:
    $$\begin{array}{rcl} \left \vert \psi _{\mathrm{reg}}(a) - \psi _{\mathrm{reg}}(b)\right \vert \leq\frac{1} {\beta }\,\max \,\left \{\psi _{\mathrm{reg}}{(a)}^{1-\beta },\psi _{\mathrm{ reg}}{(b)}^{1-\beta }\right \}{\left [\psi _{\mathrm{ reg}}(a \ast {b}^{-1})\right ]}^{\beta }\qquad & & \end{array}$$
    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
    $$\begin{array}{rcl} d_{\psi ,\beta }^{\mathrm{R}} : {\mathcal{G}}^{\mathrm{R}} \rightarrow[0,+\infty ),\,\,\quad d_{ \psi ,\beta }^{\mathrm{R}}(a,b) :={ \left [\psi _{\mathrm{ reg}}(a \ast {b}^{-1})\right ]}^{\beta },\quad \forall \,(a,b) \in {\mathcal{G}}^{\mathrm{R}}.& & \end{array}$$
    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:
    $$\begin{array}{rcl} & & \mathit{for\ any}\ (a,b) \in {\mathcal{G}}^{\mathrm{R}},\ \mathit{one\ has}\ d_{ \psi ,\beta }^{\mathrm{R}}(a,b) = 0\ \mathit{if \ and\ only\ if }\ a = b, \\ & & d_{\psi ,\beta }^{\mathrm{R}}(a,b) = d_{ \psi ,\beta }^{\mathrm{R}}(b,a)\ \mathit{for\ every}\ (a,b) \in {\mathcal{G}}^{\mathrm{R}}, \\ & & d_{\psi ,\beta }^{\mathrm{R}}(a,b) \leq d_{ \psi ,\beta }^{\mathrm{R}}(a,c) + d_{ \psi ,\beta }^{\mathrm{R}}(c,b)\ \mathit{for\ all}\ (a,c),(c,b) \in {\mathcal{G}}^{\mathrm{R}}. \end{array}$$
    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
    $$\begin{array}{c} (a,b) \in {\mathcal{G}}^{\mathrm{R}}\text{ and}\ c \in G\text{ such that }(a,c),(b,c) \in{G}^{(2)} \\ \Longrightarrow\,(a {\ast} c,b {\ast} c) \in {\mathcal{G}}^{\mathrm{R}}\text{ and }d_{ \psi ,\beta }^{\mathrm{R}}(a {\ast} c,b {\ast} c) = d_{ \psi ,\beta }^{\mathrm{R}}(a,b)\end{array}$$
    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
$$\begin{array}{rcl} \frac{d} {d +\min \,\{ d,{[\mathrm{log}_{2}C_{\rho }]}^{-1}\}} < p \leq1,& &\end{array}$$
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.


  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.



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.


  1. 1.
    L.V. Ahlfors, Bounded analytic functions. Duke Math. J. 14, 1–11 (1947)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    H. Aimar, B. Iaffei, L. Nitti, On the Macías-Segovia metrization theorem of quasi-metric spaces. Revista U. Mat. Argentina 41, 67–75 (1998)MathSciNetMATHGoogle Scholar
  3. 3.
    F. Albiac, N.J. Kalton, Lipschitz structure of quasi-Banach spaces. Israel J. Math. 170, 317–335 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    P. Alexandroff, P. Urysohn, Une condition nécessaire et suffisante pour qu’une classe (L) soit une classe (D). C. R. Acad. Sci. Paris 177, 1274–1277 (1923)Google Scholar
  5. 5.
    R. Alvarado, D. Mitrea, I. Mitrea, M. Mitrea, Weighted mixed-normed spaces on quasi-metric spaces, preprint (2012)Google Scholar
  6. 6.
    I. Amemiya, A generalization of Riesz-Fischer’s theorem. J. Math. Soc. Jpn. 5, 353–354 (1953)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    T. Aoki, Locally bounded topological spaces. Proc. Jpn. Acad. Tokyo 18, 588–594 (1942)MATHCrossRefGoogle Scholar
  8. 8.
    N. Aronszajn, Quelques remarques sur les relations entre les notions d’écart régulier et de distance. Bull. Am. Math. Soc. 44, 653–657 (1938)MathSciNetCrossRefGoogle Scholar
  9. 9.
    P. Assouad, Espaces métriques, plongements, facteurs. Thèse de doctorat d’État, Orsay, 1977MATHGoogle Scholar
  10. 10.
    P. Assouad, Étude d’une dimension métrique liée à la possibilité de plongements dans \({\mathbb{R}}^{n}\). C. R. Acad. Sci. Paris, Série A 288, 731–734 (1979)Google Scholar
  11. 11.
    P. Assouad, Plongements Lipschitziens dans \({\mathbb{R}}^{n}\). Bull. Soc. Math. France 111, 429–448 (1983)MathSciNetMATHGoogle Scholar
  12. 12.
    S. Banach, Metrische Gruppen. Studia Math. 3, 101–113 (1931)Google Scholar
  13. 13.
    S. Banach, Théorie des Opérations Linéaires, Warsaw, 1932Google Scholar
  14. 14.
    A. Benedek, R. Panzone, The space L P, with mixed norm. Duke Math. J. 28(3), 301–324 (1961)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    C. Bennett, R. Sharpley, Interpolation of operators. Pure and Applied Mathematics, vol. 129 (Academic, New York, 1988)Google Scholar
  16. 16.
    J. Bergh, J. Löfström, Interpolation Spaces. An Introduction (Springer, Berlin, 1976)Google Scholar
  17. 17.
    A.S. Besicovitch, I.J. Schoenberg, On Jordan arcs and Lipschitz classes of functions defined on them. Acta Math. 106, 113–136 (1961)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    R.H. Bing, Metrization of topological spaces. Can. J. Math. 3, 175–186 (1951)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    G. Birkhoff, A note on topological groups. Compositio Math. 3, 427–430 (1956)MathSciNetGoogle Scholar
  20. 20.
    N. Bourbaki, Topologie générale, Chapitre 9. Utilisation des nombres réels en topologie générale (Act. Sci. Ind. 1045) (Hermann, Paris, 1958)Google Scholar
  21. 21.
    H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes. Math. Annalen 96, 360–366 (1926)CrossRefGoogle Scholar
  22. 22.
    L.G. Brown, Note on the open mapping theorem. Pac. J. Math. 38(1), 25–28 (1971)MATHCrossRefGoogle Scholar
  23. 23.
    R. Brown, From groups to groupoids: a brief survey. Bull. Lond. Math. Soc. 19, 113–134 (1987)MATHCrossRefGoogle Scholar
  24. 24.
    R. Brown, Topology and Groupoids (BookSurge Publishing, 2006)Google Scholar
  25. 25.
    R.H. Bruck, A Survey of Binary Systems (Ergebnisse der Mathematik N.F. 20) (Springer, Berlin, 1958)Google Scholar
  26. 26.
    Y. Brudnyĭ, N. Krugljak, Interpolation Functors and Interpolation Spaces, vol. I (North-Holland, Amsterdam, 1991)MATHGoogle Scholar
  27. 27.
    D. Burago, Y. Burago, S.V. Ivanov, A Course in Metric Geometry (American Mathematical Society, Providence, 2001)MATHGoogle Scholar
  28. 28.
    F. Cabello Sánchez, J.M.F. Castillo, Banach space techniques underpinning a theory for nearly additive mappings, Dissertationes Math. (Rozprawy Mat.), vol. 404, 2002Google Scholar
  29. 29.
    J. Cerdà, J. Martín, P. Silvestre, Capacitary function spaces. Collect. Math. 62(1), 95–118 (2011)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    E.W. Chittenden, On the equivalence of écart and voisinage. Trans. Am. Math. Soc. 18, 161–166 (1917)MathSciNetMATHGoogle Scholar
  31. 31.
    E.W. Chittenden, On the metrization problem and related problems in the theory of abstract sets. Bull. Am. Math. Soc. 33, 13–34 (1927)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    M. Christ, in Lectures on Singular Integral Operators. CBMS Regional Conference Series in Mathematics, vol. 77 (American Mathematical Society, Providence, 1990)Google Scholar
  33. 33.
    R.R. Coifman, Y. Meyer, E.M. Stein, Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62, 304–335 (1985)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    R.R. Coifman, G. Weiss, in Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes. Lecture Notes in Mathematics, vol. 242 (Springer, Berlin, 1971)Google Scholar
  35. 35.
    R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4) 569–645 (1977)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups. Proc. Am. Math. Soc. 83, 69–70 (1981)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    G. David, J.L. Journé, S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Math. Iberoam. 1, 1–56 (1985)MATHCrossRefGoogle Scholar
  38. 38.
    G. David, S. Semmes, in Fractured Fractals and Broken Dreams: Self-similar Geometry Through Metric and Measure. Oxford Lecture Series in Mathematics and its Applications, vol. 7 (Clarendon, Oxford University Press, New York, 1997)Google Scholar
  39. 39.
    D. Deng, Y. Han, in Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, vol. 1966 (Springer, Berlin, 2009)Google Scholar
  40. 40.
    A. Di Concilio, S.A. Naimpally, A unified approach to metrization problems. Acta Math. Hungarica 53(1–2), 109–113 (1998)Google Scholar
  41. 41.
    J. Dieudonné, L. Schwartz, La dualité dans les espaces (F) et (LF). Ann. Inst. Fourier, Grenoble 1 (1949), 61–101 (1950)Google Scholar
  42. 42.
    J.J. Dudziak, Vitushkin’s Conjecture for Removable Sets (Universitext) (Springer, Berlin, 2010)Google Scholar
  43. 43.
    V.A. Efremovič, A.S. Švarc, A new definition of uniform spaces. Metrization of proximity spaces, (Russian) Doklady Akad. Nauk SSSR (N.S.) 89, 393–396 (1953)Google Scholar
  44. 44.
    R. Engelking, General Topology (Heldermann, Berlin, 1989)MATHGoogle Scholar
  45. 45.
    G.B. Folland, E. Stein, Hardy Spaces on Homogeneous Groups (Princeton University Press, Princeton, 1982)MATHGoogle Scholar
  46. 46.
    M. Frazier, B. Jawerth, Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    M. Fréchet, Les dimensions d’une ensemble abstrait. Math. Ann. 68, 145–168 (1909–1910)Google Scholar
  49. 49.
    A.H. Frink, Distance functions and the metrization problem. Bull. Am. Math. Soc. 43, 133–142 (1937)MathSciNetCrossRefGoogle Scholar
  50. 50.
    I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, in Weighted Theory for Integral Transforms on Spaces of Homogeneous Type. Pitman Monographs and Surveys in Pure and Applied Mathematics, Addison Wesley Longman Inc. vol. 92 (1998)Google Scholar
  51. 51.
    A. Gogatishvili, P. Koskela, N. Shanmugalingam, in Interpolation Properties of Besov Spaces Defined on Metric Spaces. Mathematische Nachrichten, Special Issue: Erhard Schmidt Memorial Issue, Part II, vol. 283, Issue 2 (2010), pp. 215–231Google Scholar
  52. 52.
    A. Grothendieck, in Produits Tensoriels Topologique et Espaces Nucléaires. Memoirs of the American Mathematical Society, vol. 16 (AMS, Providence, 1955)Google Scholar
  53. 53.
    J. Gustavsson, Metrization of quasi-metric spaces. Math. Scand. 35, 56–60 (1974)MathSciNetMATHGoogle Scholar
  54. 54.
    P. Hajłasz, Whitney’s example by way of Assouad’s embedding. Proc. Am. Math. Soc. 131(11), 3463–3467 (2003)MATHCrossRefGoogle Scholar
  55. 55.
    Y. Han, D. Müller, D. Yang, A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal. no. 893409, 1–250 (2008)CrossRefGoogle Scholar
  56. 56.
    Y. Han, E. Sawyer, in Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces. Memoirs of the American Mathematical Society, vol. 530 (AMS, Providence, 1994)Google Scholar
  57. 57.
    F. Hausdorff, Grundzüge der Mengenlehre (Von Veit, Leipzig, 1914)MATHGoogle Scholar
  58. 58.
    W. Hebisch, A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group. Studia Math. 96(3), 231–236 (1990)MathSciNetMATHGoogle Scholar
  59. 59.
    J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext (Springer, New York, 2001)Google Scholar
  60. 60.
    T. Holmstedt, Interpolation of quasi-normed spaces. Math. Scand. 26, 177–199 (1970)MathSciNetMATHGoogle Scholar
  61. 61.
    L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. I (reprint of the 2-nd edition 1990) (Springer, Berlin, 2003)Google Scholar
  62. 62.
    G. Hu, D. Yang, Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwanese J. Math. 133(1), 91–135 (2009)MathSciNetGoogle Scholar
  63. 63.
    T. Husain, S-spaces and the open mapping theorem. Pac. J. Math. 12(1), 253–271 (1962)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    T. Husain, Introduction to Topological Groups (W.B. Saunders, Philadelphia, 1966)MATHGoogle Scholar
  65. 65.
    S. Kakutani, Über die Metrisation der topologischen Gruppen. Proc. Imp. Acad. Jpn. 12, 82–84 (1936)MathSciNetCrossRefGoogle Scholar
  66. 66.
    N.J. Kalton, Basic sequences in F-spaces and their applications. Proc. Edinb. Math. Soc. (2) 19(2), 151–167 (1974/1975)Google Scholar
  67. 67.
    N.J. Kalton, in Quasi-Banach spaces, ed. by W.B. Johnson, J. Lindenstrauss. Handbook of the Geometry of Banach Spaces. Chapter 25 in vol. 2 Elsevier Science B. V. (2003)Google Scholar
  68. 68.
    N. Kalton, S. Mayboroda, M. Mitrea, in Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin Spaces and Applications to Problems in Partial Differential Equations, ed. by L. De Carli, M. Milman. Interpolation Theory and Applications. Contemporary Mathematics, vol. 445 (American Mathematical Society, Providence, 2007), pp. 121–177Google Scholar
  69. 69.
    N.J. Kalton, N.T. Peck, J.W. Roberts, in An F-space Sampler. London Mathematical Society Lecture Notes Series, vol. 89 (Cambridge University Press, Cambridge, 1984)Google Scholar
  70. 70.
    A. Kamińska, Some remarks on Orlicz-Lorentz spaces. Math. Nachr. 147, 29–38 (1990)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    J.L. Kelley, General Topology (van Nostrand, Toronto, 1955)Google Scholar
  72. 72.
    M.D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen. Fund. Math. 22, 77–108 (1934)Google Scholar
  73. 73.
    P. Koskela, N. Shanmugalingam, H. Tuominen, Removable sets for the Poincaré inequality on metric spaces. Indiana Math. J. 49, 333–352 (2000)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    G. Köthe, Topological Vector Spaces I (Springer, Berlin, 1969)MATHCrossRefGoogle Scholar
  75. 75.
    C. Kuratowski, Quelques problèmes concernant les espaces métriques non-séparables. Fund. Math. 25, 534–545 (1935)Google Scholar
  76. 76.
    S. Leader, Metrization of proximity spaces. Proc. Am. Math. Soc. 18, 1084–1088 (1967)MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    J. Luukkainen, H. Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces. Fund. Math. 144, 181–193 (1994)MathSciNetMATHGoogle Scholar
  78. 78.
    J. Luukkainen, E. Saksman, Every complete doubling metric space carries a doubling measure. Proc. Am. Math. Soc. 126(2), 531–534 (1998)MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    R.A. Macías, C. Segovia, Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)MATHCrossRefGoogle Scholar
  80. 80.
    R.A. Macías, C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type. Adv. Math. 33(3), 271–309 (1979)MATHCrossRefGoogle Scholar
  81. 81.
    E.J. McShane, Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934)MathSciNetCrossRefGoogle Scholar
  82. 82.
    D. Mitrea, I. Mitrea, M. Mitrea, E. Ziadé, Abstract capacitary estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces. J. Funct. Anal. 262, 4766–4830 (2012)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    I. Mitrea, M. Mitrea, E. Ziadé, in A quantitative Open Mapping Theorem for quasi-pseudonormed groups, Advances in Harmonic Analysis and Applications, a volume in honor of K.I. Oskolkov, Springer Proceedings in Mathematics, 25, 259–286 (2013)Google Scholar
  84. 84.
    D. Montgomery, L. Zippin, Topological Transformation Groups (Interscience Publishers, New York, 1955)MATHGoogle Scholar
  85. 85.
    S. Montgomery-Smith, in Boyd indices of Orlicz-Lorentz spaces. Function Spaces (Edwardsville, IL, 1994). Lecture Notes in Pure and Applied Mathematics, vol. 172 (Dekker, New York, 1995), pp. 321–334Google Scholar
  86. 86.
    P.S. Muhly, Coordinates in Operator Algebras, book manuscript (1997)Google Scholar
  87. 87.
    J.R. Munkres, Topology, 2nd edn. (Prentice Hall, Englewood Cliffs, NJ, 2000)MATHGoogle Scholar
  88. 88.
    J. Nagata, On a necessary and sufficient condition of metrizability. J. Inst. Polytech. Osaka City Univ. Ser. A. Math. 1, 93–100 (1950)MathSciNetGoogle Scholar
  89. 89.
    F. Nazarov, S. Treil, A. Vol’berg, Tb-theorem on non-homogeneous spaces. Acta Math. 190(2), 151–239 (2003)Google Scholar
  90. 90.
    V.W. Niemytzki, On the third axiom of metric spaces. Trans. Am. Math. Soc. 29, 507–513 (1927)MathSciNetMATHGoogle Scholar
  91. 91.
    S. Okada, W.J. Ricker, E.A. Sánchez Pérez, in Optimal Domain and Integral Extension of Operators. Operator Theory, Advances and Applications, vol. 180 (Birkhäuser, Basel, 2008)Google Scholar
  92. 92.
    J.C. Oxtoby, Cartesian products of Baire spaces. Fund. Math. 49, 157–166 (1961)MathSciNetMATHGoogle Scholar
  93. 93.
    M. Paluszyński, K. Stempak, On quasi-metric and metric spaces. Proc. Am. Math. Soc. 137, 4307–4312 (2009)MATHCrossRefGoogle Scholar
  94. 94.
    P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. 129, 1–60 (1989)MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    A.R. Pears, Dimension Theory of General Spaces (Cambridge University Press, London, 1975)MATHGoogle Scholar
  96. 96.
    J. Peetre, Espaces d’interpolation, généralisations, applications. Rend. Sem. Mat. Fis. Milano 34, 133–164 (1964)MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    J. Peetre, G. Sparr, Interpolation of normed Abelian groups. Ann. Math. Pura Appl. 92, 217–262 (1972)MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    B. Pettis, On continuity and openness of homomorphisms in topological groups. Ann. Math. 54, 293–308 (1950)MathSciNetCrossRefGoogle Scholar
  99. 99.
    A. Pietsch, History of Banach Spaces and Linear Operators (Birkhäuser, Boston, 2007)MATHGoogle Scholar
  100. 100.
    D. Pompeiu, Sur la continuité des fonctions de variables complexes (Thèse), Gauthier-Villars, Paris, 1905; Ann. Fac. Sci. de Toulouse 7, 264–315 (1905)MathSciNetMATHGoogle Scholar
  101. 101.
    J. Renault, in A Groupoid Approach to C -Algebras. Lecture Notes in Mathematics, vol. 793 (Springer, Berlin, 1980)Google Scholar
  102. 102.
    A.P. Robertson, W. Robertson, On the closed graph theorem. Proc. Glasgow Math. Ass. 3, 9–12 (1956)MATHCrossRefGoogle Scholar
  103. 103.
    S. Rolewicz, On a certain class of linear metric spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 5, 471–473 (1957)MathSciNetMATHGoogle Scholar
  104. 104.
    S. Rolewicz, Metric Linear Spaces (D. Reidel, Dordrecht, 1985)MATHGoogle Scholar
  105. 105.
    D. Rolfsen, Alternative metrization proofs. Can. J. Math. 18, 750–757 (1966)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    H.L. Royden, Real Analysis, 2nd edn. (MacMillan, New York, 1968)Google Scholar
  107. 107.
    W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1976)Google Scholar
  108. 108.
    W. Rudin, in Functional Analysis, 2nd edn. International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991)Google Scholar
  109. 109.
    T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators (de Gruyter, Berlin, 1996)Google Scholar
  110. 110.
    S. Semmes, Bilipschitz embeddings of metric spaces into Euclidean spaces. Publ. Math. 43(2), 571–653 (1999)MathSciNetMATHGoogle Scholar
  111. 111.
    R. Sikorski, Boolean Algebras (Springer, Berlin, 1960)MATHGoogle Scholar
  112. 112.
    Y. Smirnov, A necessary and sufficient condition for metrizability of a topological space. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 77, 197–200 (1951)Google Scholar
  113. 113.
    E.M. Stein, in Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30 (Princeton University Press, Princeton, 1970)Google Scholar
  114. 114.
    E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, 1993)MATHGoogle Scholar
  115. 115.
    A.H. Stone, Sequences of coverings. Pac. J. Math. 10, 689–691 (1960)MATHGoogle Scholar
  116. 116.
    X. Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math. 190, 105–149 (2003)MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    X. Tolsa, Analytic capacity, rectifiability, and the Cauchy integral, in Proceedings of the ICM, Madrid, 2006, pp. 1505–1527Google Scholar
  118. 118.
    A. Torchinsky, Interpolation of operators and Orlicz classes. Studia Math. 59, 177–207 (1976)MathSciNetMATHGoogle Scholar
  119. 119.
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd revised and enlarged edition (Johann Ambrosius Barth, Heidelberg, 1995)Google Scholar
  120. 120.
    H. Triebel, Theory of Function Spaces (Birkhäuser, Berlin, 1983)CrossRefGoogle Scholar
  121. 121.
    H. Triebel, in Theory of Function Spaces, II. Monographs in Mathematics, vol. 84 (Birkhäuser, Basel, 1992)Google Scholar
  122. 122.
    H. Triebel, A new approach to function spaces on spaces of homogeneous type. Rev. Mat. Comput. 18(1), 7–48 (2005)MathSciNetMATHGoogle Scholar
  123. 123.
    H. Triebel, Theory of Function Spaces III (Birkhäuser, Basel, 2006)MATHGoogle Scholar
  124. 124.
    A. Tychonoff, Über einen Metrisationssatz von P. Urysohn. Math. Ann. 95, 139–142 (1926)MathSciNetCrossRefGoogle Scholar
  125. 125.
    P. Urysohn, Zum Metrisationsproblem. Math. Ann. 94, 309–315 (1925)MathSciNetMATHGoogle Scholar
  126. 126.
    D.A. Vladimirov, in Boolean Algebras in Analysis. Mathematics and Its Applications (Kluwer, Dordrecht, 2002)Google Scholar
  127. 127.
    A.L. Vol’berg, S.V. Konyagin, On measures with the doubling condition. Izv. Akad. Nauk SSSR Ser. Mat. 51(3), 666–675 (1987) (Russian); translation in Math. USSR-Izv., 30(3), 629–638 (1988)Google Scholar
  128. 128.
    A. Weil, Sur les espaces à structure uniforme et sur la topologie générale. Act. Sci. Ind. Paris 551 (1937)Google Scholar
  129. 129.
    H. Whitney, Analytic extensions of functions defined on closed sets. Trans. Am. Math. Soc. 36, 63–89 (1934)MathSciNetCrossRefGoogle Scholar
  130. 130.
    A.C. Zaanen, Integration (North-Holland, Amsterdam, 1967)MATHGoogle Scholar

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

Personalised recommendations