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2001 | Buch

Covering Theorems Kapitel lesen Erstes Kapitel lesen

Lectures on Analysis on Metric Spaces

verfasst von: Juha Heinonen

Verlag: Springer New York

Buchreihe : Universitext

Enthalten in: Professional Book Archive

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Über dieses Buch

Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.

Inhaltsverzeichnis

Frontmatter
1. Covering Theorems
Abstract
All covering theorems are based on the same idea: from an arbitrary cover of a set in a metric space, one tries to select a subcover that is, in a certain sense, as disjointed as possible. In order to have such a result, one needs to assume that the covering sets are somehow nice, usually balls. In applications, the metric space normally comes with a measure μ, so that if F = {B} is a covering of a set A by balls, then always
$$ \mu (A) \leqslant \sum\limits_\mathcal{F} {\mu (B)} $$
(with proper interpretation of the sum if the collection F is not countable). What we often would like to have, for instance, is an inequality in the other direction,
$$ \mu (A) \geqslant C \sum\limits_{\mathcal{F}'} {\mu (B)} , $$
for some subcollection F′ ⊂ F that still covers A and for some positive constant C that is independent of A and the covering F. There are many versions of this theme.
Juha Heinonen
2. Maximal Functions
Abstract
Throughout this chapter, (X, μ) is a doubling metric measure space.
Juha Heinonen
3. Sobolev Spaces
Abstract
We denote by ℝ n Euclidean n-space, n ≥ 1, and by dx its Lebesgue measure.
Juha Heinonen
4. Poincaré Inequality
Abstract
The Sobolev inequality
$$ \left\| u \right\|_{np/(n - p)} \leqslant C (n,p)\left\| {\nabla u} \right\|_p $$
(4.1)
for 1 ≤ p < n cannot hold for an arbitrary smooth function u that is defined only, say, in a ball B. For instance, if u is a nonzero constant, the right-hand side is zero but the left-hand side is not. However, if we replace the integrand on the left-hand side by ‖u − u B ‖, where, we recall, u B is the mean value of u in the ball B, an appropriate form of inequality (4.1) is salvaged: the inequality
$$ \left( {\int {_B \left| {u - u_B } \right|^{np/(n - p)} dx} } \right)^{(n - p)/np} \leqslant C(n,p)\left( {\int {_B \left| {\nabla u} \right|^p dx} } \right)^{1/p} $$
(4.2)
holds for all smooth functions u in a ball B in ℝ n , if 1 ≤ p < n. Consequently, inequality (4.2) holds for all functions u in the Sobolev space W1,p(B). Inequality (4.2) is often called the Sobolev-Poincaré inequality, and it will be proved momentarily. Before that, let us derive a weaker inequality (4.4) from inequality (4.2) as follows. By inserting the measure of the ball B into the integrals, we find that
$$ \left( {f_B \left| {u - u_B } \right|^{np/(n - p)} dx} \right)^{(n - p)/np} \leqslant C(n,p)(diam B)\left( {f_B \left| {\nabla u} \right|^p dx} \right)^{1/p} $$
(4.3)
and hence, by Hölder’s inequality, that
$$ f_B \left| {u - u_B } \right|^p dx \leqslant C (n,p)(diam B)^p f_B \left| {\nabla u} \right|^p dx, $$
(4.4)
which is customarily known as the Poincaré inequality. In fact, inequality (4.4) is valid for all 1 ≤ p < ∞.
Juha Heinonen
5. Sobolev Spaces on Metric Spaces
Abstract
We have seen that if u is a smooth function defined on a ball B in ℝ n (possibly with infinite radius so that B = ℝ n ), then the inequality
$$ \left| {u(x) - u(y)} \right| \leqslant C (n)(I_1 \left| {\nabla u} \right|(x) + I_1 \left| {\nabla u} \right|(y)) $$
(5.1)
holds for each pair of points x, y in B, where I1 is the Riesz potential. It is easily seen that in the definition of I1 we can integrate ‖∇u‖ against the Riesz kernel ‖z‖1−n over a ball whose radius is roughly ‖x − y‖ and still retain inequality (5.1); then, I1‖∇u‖(x) is controlled by a constant C(n) times
$$ \left| {x - y} \right|M\left| {\nabla u} \right|(x), $$
as is easily seen by dividing the ball over which the integration occurs into annuli as in the proof of Proposition 3.19. By symmetry, there is a similar bound for I1‖∇u‖(y), and we therefore conclude that
$$ \left| {u(x) - u(y)} \right| \leqslant C(n)\left| {x - y} \right|(M\left| {\nabla u} \right|(x) + M\left| {\nabla u} \right|(y)) $$
(5.2)
for each pair of points x, y in B. If u belongs to W1,p(B), so that its gradient is in L p (B), and if p > 1, we conclude from the maximal function theorem that
$$ \left| {u(x) - u(y)} \right| \leqslant \left| {x - y} \right|(g(x) + g(y)) $$
(5.3)
for each pair of points x, y in B, where g ε L p (B); in fact, we can choose g in inequality (5.3) to be a constant times the maximal function M‖∇u‖. By the density of smooth functions in W1,p(B), we thus obtain that inequality (5.3) continues to hold almost everywhere in B (in the sense that by ruling out a set of measure zero, inequality (5.3) holds for all points x and y outside this set).
Juha Heinonen
6. Lipschitz Functions
Abstract
Lipschitz functions are the smooth functions of metric spaces. A real-valued function f on a metric space X is said to be L-Lipschitz if there is a constant L ≥ 1 such that
$$ \left| {f(x) - f(y)} \right| \leqslant L\left| {x - y} \right| $$
(6.1)
for all x and y in X. Of course, there is nothing special about having the real line as a target, and in general we call a map f : X → Y between metric spaces Lipschitz, or L-Lipschitz if the constant L ≥ 1 deserves to be mentioned, if condition (6.1) holds.
Juha Heinonen
7. Modulus of a Curve Family, Capacity, and Upper Gradients
Abstract
In Chapter 5, we discussed a possible definition for a Sobolev space in a metric measure space; this was the space M1,p of Hajłasz. While often convenient, it does not capture the geometry of the underlying space. This is seen, for example, in the fact that a Poincaré inequality always holds for functions in M1,p (Theorem 5.15). On the other hand, the validity of a Sobolev-Poincaré-type inequality in a space should tell us something about the geometry of the space, as in the discussion of the isoperimetric inequalities in Section 3.30. Now M1,p fails in this task because the definition of the “derivative” g of a function u in M1,p already is global. There is necessarily a loss of information. (In some sense, M1,p precisely consists of those functions for which a Poincaré inequality is satisfied [75].)
Juha Heinonen
8. Loewner Spaces
Abstract
Let (X, μ) be a metric measure space. For each real number n > 1, we define the Loewner function φ n : (0, ∞) → [0, ∞) of X by
$$ \varphi _{X,n} (t) = \varphi _n (t) = \inf \{ \bmod _n (E, F; X):\Delta (E, F) \leqslant t\} , $$
where E and F are disjoint nondegenerate continua in X with
$$ \Delta (E, F) = \frac{{dist(E,F)}} {{\min \{ diam E, diam F\} }} $$
designating their relative position in X. Here (E, F; X) denotes the family of all curves joining E and F in X. If X is understood from the context, we usually write (E, F; X) = (E, F). If one cannot find two disjoint continua in X, it is understood that φ X,n (t) ≡ 0. Recall that a continuum is a compact connected set, and a continuum is nondegenerate if it is not a point; henceforth, we tacitly assume that all continua are nondegenerate.
Juha Heinonen
9. Loewner Spaces and Poincaré Inequalities
Abstract
In Chapter 4, we proved the Poincaré inequality in ℝ n by using the fact that points in ℝ n can be joined by a thick “pencil” of curves. In the previous chapter, we defined the Loewner function of a metric space that detects quantitatively whether or not the space contains rectifiable curves. In this chapter, we will see that these two concepts, the Poincaré inequality and the Loewner condition, are related.
Juha Heinonen
10. Quasisymmetric Maps: Basic Theory I
Abstract
In this chapter, we develop a basic theory of quasisymmetric embeddings in metric spaces, following for the most part the paper by Tukia and Väisälä [176]. We use the notation f : X → Y for an embedding f of a metric space X in a metric space Y. Thus, note in particular that in this notation f is not supposed to be onto. Recall that an embedding is a map that is a homeomorphism onto its image.
Juha Heinonen
11. Quasisymmetric Maps: Basic Theory II
Abstract
In this chapter, we first introduce an important quasisymmetrically invariant class of metric spaces and then prove that quasisymmetric maps between spaces from this class are Hölder continuous. Quantitative bounds for the change in Hausdorff dimension then follow. As a second topic, we discuss the relationship between quasisymmetry and quasiconformality for maps between Euclidean domains. Finally, as an example, we show how quasisymmetric maps naturally arise in one-variable complex dynamics.
Juha Heinonen
12. Quasisymmetric Embeddings of Metric Spaces in Euclidean Space
Abstract
As in topology, where one wants to understand the homeomorphism type of a given space, a basic question in the theory of quasisymmetric maps asks which metric spaces are quasisymmetrically homeomorphic. This question is extremely difficult in general. There are closed (compact without boundary) four-dimensional Riemannian manifolds that are homeomorphic but not quasisymmetrically homeomorphic [42]. An easier question asks which spaces can be embedded quasisymmetrically in a given space or in a space from a given collection. A beautiful and complete answer to this problem can be given in the case where the receiving space is Euclidean.
Juha Heinonen
13. Existence of Doubling Measures
Abstract
If μ is a doubling measure on a metric space X, then it is easy to see that X is doubling as defined in Section 10.13. On the other hand, not every doubling space carries a doubling measure.
Juha Heinonen
14. Doubling Measures and Quasisymmetric Maps
Abstract
In this chapter, we further explore the close connection between doubling measures and quasisymmetric maps. The main theme is that doubling measures can be used to deform the underlying metric space and that such deformations are quasisymmetric. This point of view has been advocated by David and Semmes; see [39], [164], and [165].
Juha Heinonen
15. Conformal Gauges
Abstract
In this final chapter, we introduce a convenient language to discuss quasisymmetrically invariant properties of metric spaces.
Juha Heinonen
Backmatter
Metadaten
Titel
Lectures on Analysis on Metric Spaces
verfasst von
Juha Heinonen
Copyright-Jahr
2001
Verlag
Springer New York
Electronic ISBN
978-1-4613-0131-8
Print ISBN
978-1-4612-6525-2
DOI
https://doi.org/10.1007/978-1-4613-0131-8