Dimension distortion by Sobolev mappings in foliated metric spaces
We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space defined by a David–Semmes regular mapping , we quantitatively estimate, in terms of Hausdorff dimension in , the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.
Key words and phrases:Sobolev mapping, Ahlfors regularity, Poincaré inequality, foliation, David–Semmes regular mapping
2010 Mathematics subject classification. Primary: 46E35, 28A78; Secondary: 46E40, 53C17, 30L99
Let and be differentiable manifolds of dimensions . For every , the preimage under a submersion is a submanifold of of dimension . In this way, the map defines a foliation of parameterized by . The canonical such submersion is the orthogonal projection of onto the codimension one subspace spanned by all but the coordinate vector, . The resulting foliation of by parallel straight lines features in the theory of Sobolev mappings. Indeed, a mapping is in the Sobolev space if and only if, up to choice of representative, for each , each coordinate function of is absolutely continuous on -almost every line in the above foliation, and the resulting partial derivatives are in .
Sobolev mappings between metric spaces are of growing interest and importance in modern analysis, geometric group theory, and geometric measure theory. While there are several (often equivalent) definitions of such Sobolev maps, in each approach Sobolev mappings are assumed or shown to be absolutely continuous along “almost every curve”. When the space under consideration is equipped with a foliation by curves that is parameterized by another space , it is natural that “almost every curve” refer to a measure on , as in the Euclidean case above. In other words, a Sobolev mapping should preserve or decrease the dimension of almost every leaf of a given foliation by curves.
Our first result states that while a Sobolev mapping may substantially increase the dimension of the remaining measure zero set of curves, this increase is controlled. In fact, there are universal bounds on the dimension increase under a supercritical Sobolev mapping. Here and henceforth in this paper we denote by the -dimensional Hausdorff measure in a metric space , and by the Hausdorff dimension of a subset of the space . The assumptions on the metric space in Theorem 1.1 are standard and explained in Section 2 below, as is the notion of a Sobolev space based on upper gradients.
Let be a proper metric measure space that is locally -homogeneous and supports a local -Poincaré inequality. Let be any metric space. For , if is a continuous mapping that has an upper gradient in , then
for any subset . Moreover, if , then .
In the setting of quasiconformal mappings between domains in Euclidean space, such dimension distortion estimates have been known for many years; see, for instance, Gehring , Gehring–Väisälä  and Astala . Theorem 1.1 was also known in the Euclidean Sobolev setting; see for instance Kaufman [23, Theorem 1].
The main difficulty in proving Theorem 1.1 is that in the general metric setting, the usual analogue of a Euclidean dyadic cube need not be bi-Lipschitz equivalent to a ball of a comparable diameter. Hence, nice essentially disjoint coverings of sets need not exist. We overcome this obstacle by using maximal functions; see Lemma 3.3. In the case that the set under consideration in Theorem 1.1 is Ahlfors regular, we reach the stronger conclusion that has zero measure in the appropriate dimension; see Theorem 4.1 below.
We also prove that Theorem 1.1 is sharp whenever the domain is Ahlfors regular, and that the collection of Sobolev mappings which increase the dimension of a given compact set by the maximum possible amount is prevalent, a notion of genericity in Banach spaces (see , , , , ). Here the relevant Banach space is a Newtonian–Sobolev space, defined in Section 3.
Let be a metric measure space that is locally Ahlfors -regular for some . For and , set
Let be a compact subset of such that . Then for all greater than , there exists a continuous map such that has an upper gradient in and . Moreover, the set of such functions forms a prevalent set in the Newtonian–Sobolev space .
We now turn to the study of dimension increase under Sobolev mappings of generic leaves in parameterized families of subsets. Consider a continuous supercritical Sobolev mapping as in Theorem 1.1 and fix a dimension and a target dimension . If is foliated by subsets of dimension no greater than , how many leaves of the foliation can be mapped by onto sets of dimension at least ? The answer will be given in terms of Hausdorff measures on the parameterizing space for the foliation.
The preceding question has been thoroughly studied in Euclidean space. The first two authors together with Monti  studied supercritical () and borderline () Sobolev mappings on foliations arising from the orthogonal projection of onto a subspace of arbitrary dimension. In the same setting, Hencl and Honzík  considered the sub-critical case (). Bishop and Hakobyan  have recently addressed finer questions for the behavior of planar quasiconformal mappings along lines.
In this general metric setting, we must first give precise meaning to the notion of foliation. David and Semmes  introduced a class of mappings between metric spaces that is analogous to the class of submersions between differentiable manifolds. In this paper, we study foliations arising from local versions of David–Semmes regular mappings.
Let . A surjection between proper metric spaces is said to be locally David–Semmes -regular (for short, locally -regular) if for every compact subset , is Lipschitz and there is a constant and a radius such that for every ball of radius , the truncated preimage can be covered by at most balls in of radius .
An easy calculation shows that given a locally -regular mapping , a compact subset , and a point ,
where depends only on the constant associated to in Definition 1.3. In particular, the leaves have Hausdorff dimension no greater than . However, leaves can have Hausdorff dimension strictly less than ; this situation occurs naturally for certain foliations of the Heisenberg group, as we will see later.
Let be a locally -regular mapping. The triple will be called an -foliation of . If the value of is unimportant, we will refer to an -foliation as a metric foliation. Given a point , the set is called a leaf of the foliation.
Let and . Let be a proper metric measure space that is locally -homogeneous, supports a local -Poincaré inequality, and is equipped with an -foliation . Let be any metric space. For , if is a continuous mapping that has an upper gradient in , then
for each .
The metric setting presents several obstacles to a straightforward adaptation of the Euclidean proof given in [5, Theorem 1.3]. We make heavy use of the machinery of geometric measure theory in metric spaces espoused in .
A motivating example of a metric measure space to which our results apply is the Heisenberg group . Of particular interest are foliations of by either left or right cosets of a given horizontal subgroup. When the leaves are left cosets, they are also horizontal, and the natural parameterizing space is Euclidean. However, when the leaves are right cosets, they are rarely horizontal, and the natural parameterizing space is the Grushin plane, a sub-Riemannian metric space homeomorphic but not bi-Lipschitz equivalent to . In this case, the wide generality of Theorem 1.4 is needed. These and other examples are discussed in Section 6.
Theorem 1.4 recovers the Euclidean result in the case of orthogonal projections onto subspaces, and is sharp in that setting [5, Theorem 1.4]. However, it is interesting to note that while the foliation of the Heisenberg group by left cosets of a horizontal subgroup is a -foliation, the dimension of a leaf is only . This prevents Theorem 1.4 from being sharp, and indicates that the framework of David-Semmes foliations is not appropriate. In the article , we provide an alternate framework, based on the Radon-Nikodym theorem, which is more appropriate. Notably, this alternate framework cannot accommodate foliations that are not parameterized by a Euclidean space, such as the foliation of the Heisenberg group by right cosets of a horizontal subgroup.
We now give an outline for this paper. In Section 2, we establish notation and recall relevant definitions from the theory of analysis on metric spaces. Section 3 describes the version of Morrey’s inequality, a key tool in our proofs, that is valid in the metric measure space setting. We prove Theorems 1.1 and 1.2 in Section 4. Section 5 contains the proof of Theorem 1.4. In Section 6 we provide examples of metric foliations and discuss the applications of our results. The final Section 7 contains some open questions and problems motivated by this work.
Research for this paper was conducted during a visit of the third author to the Department of Mathematics of the University of Illinois at Urbana-Champaign in January 2012, and during a visit of the second author to the Institute of Mathematics at the University of Bern in June 2012. We would like to thank these institutions for their hospitality.
2. The metric measure space setting
In a metric space , we denote the open ball centered at a point of radius by
and the corresponding closed ball by
When there is no danger of confusion, we often write in place of . A similar convention will be used for all objects that depend implicitly on the ambient space. For a subset of and a number , we denote the -neighborhood of by
For an open ball and a parameter , we set .
A metric space is proper if every closed ball is compact. We will only consider proper metric spaces in this paper.
A metric measure space is a triple where is a metric space and is a measure on . The measure is assumed to be a Borel measure that gives positive and finite value to any non-empty open set. For , we denote by the restriction of to . Given a mapping from to some other metric space , we define the push-forward measure of by as
For , the -dimensional Hausdorff measure on a metric space will be denoted by or simply . For , the corresponding pre-measure will be denoted by or , and the corresponding content will be denoted by or . Unless otherwise noted, for we denote by the Hausdorff dimension of the metric space . We refer to Mattila  for more details and information about geometric measure theory in metric spaces.
Let . We say that the metric measure space is locally -homogeneous if for every compact subset , there is a radius and a constant such that
whenever are concentric balls centered in . When the value of is unimportant, we say that is locally homogeneous.
Any locally -homogeneous space has Hausdorff dimension at most . In fact, such spaces have Assouad dimension at most ; note that the Assouad dimension is always greater than or equal to the Hausdorff dimension. We will not make use of Assouad dimension in this paper.
Every locally homogeneous metric measure space is locally doubling, which means that for every compact subset , there is a radius and a constant such that
whenever is a ball centered in with .
The local homogeneity condition only provides lower bounds on measure. We will occasionally require upper bounds as well. The metric measure space is locally Ahlfors -regular if for every compact subset , there is a radius and a constant such that
whenever is a ball centered in of radius .
We will often consider conditions on spaces and mappings defined using multiplicative constants. When estimating quantities involving such constants, we use the notation to mean that there is a constant , depending only on certain specified and fixed quantities, such that .
3. Sobolev classes and Morrey’s estimate
Let , and let . Each mapping in the supercritical Sobolev space , , has a -Hölder continuous representative satisfying Morrey’s estimate
for each ball or cube ; see, e.g., . Here denotes the norm of the matrix of weak partial derivatives of the coordinate functions of , and is a positive constant depending only on and . This fact is the sole property of Sobolev mappings needed for the results in this paper. Note that by Hölder’s inequality,
for all . Hence, if , the inequality (3.1) also holds with replaced by any exponent .
We wish to state a version of the Morrey inequality in the metric measure space setting. Throughout this section, we assume that is a proper metric measure space and that is an arbitrary metric space.
A robust approach to Sobolev spaces of mappings between metric spaces is based on the concept of an upper gradient , , . Let be a continuous map, and let be a Borel function. The function is an upper gradient of if for every rectifiable curve ,
We consider mappings which have upper gradients in . Such mappings are absolutely continuous on “most” rectifiable curves in [30, Proposition 3.1]. When the target space is a Banach space, one can define a Banach space of Newtonian-Sobolev mappings consisting of equivalence classes of (not necessarily continuous) mappings in with an upper gradient in . For more details, see  or .
If there are no rectifiable curves in , then any mapping has the zero function as an upper gradient, and so there is no hope for a Morrey estimate. The -Poincaré inequality remedies this , , .
Let . A metric measure space satisfies a local -Poincaré inequality if for every compact subset , there are constants , , and such that if is a continuous function and is an upper gradient of , then
for each open ball centered in of radius less than .
Theorem 3.2 (Hajłasz-Koskela, Heinonen et al.).
Assume that is locally -homogeneous, , and supports a local -Poincaré inequality. If is a continuous mapping with an upper gradient for some , then for each compact subset there exist constants , , and satisfying
for each open ball centered at a point of of radius less than .
In the above result, it is assumed a priori that the mapping is continuous. In fact, one could instead assume only that the mapping is locally integrable in a suitable sense; it then follows that has a continuous representative satisfying the desired conclusion.
It may occur that the quantity in Theorem 3.2 is necessarily strictly larger than one [16, Section 9]. This is an inconvenience when working with coverings. In many situations the following statement, which we learned from Koskela and Zürcher, ameliorates this problem.
Assume that is a locally doubling metric measure space and let and . For each , there is a Borel function such that for each compact set there exists a constant and a radius so that
for each and each .
Let . Then
where is a suitably restricted maximal function of [17, Chapter 2]. Integrating the above inequality over yields
The local doubling condition now implies that satisfies the requirements of the statement. ∎
Assume that is locally -homogeneous, , and supports a local -Poincaré inequality. Let . If is a continuous mapping with an upper gradient in , then there exists a Borel function such that for each compact set , there exists a constant and a radius such that
for each and .
4. Universal bounds on dimension distortion
Proof of Theorem 1.1.
We first consider the case . Fix , and choose so close to that
The countable subadditivity of Hausdorff measure allows us to assume that is contained in a ball , which has compact closure. Hence, by Proposition 3.4, there is a constant , a radius , and a Borel function such that for every and ,
Let . Since , it holds that . Hence, it follows from the definitions and the -covering theorem [17, Theorem 1.2] that there is a collection of balls centered in such that
Since is uniformly continuous on small sets, choosing small enough ensures that for all
Reducing to be less than if necessary, and using (4.1) and local homogeneity, we see that
Applications of Hölder’s inequality and the disjointness assumption now yield
Since , letting tend to zero shows that . Letting tend to now yields the desired result.
We now consider the case . Choose any . As before, we use Proposition 3.4 to find a constant , a radius , and a Borel function such that
for every and . This implies that
Since , there is a sequence of subsets of such that for each , there is a number such that
for each and , and . Then satisfies , and
Note that local -homogeneity implies that is absolutely continuous with respect to . Hence . It suffices to show that as well; the proof of this is analogous to the proof in the previous case and is left to the reader. This also proves the final statement of the Theorem. ∎
In order to reach the stronger conclusion that the image of a given set has zero measure in the appropriate dimension, we assume the set has additional structure:
Assume the notation and hypotheses of Theorem 1.1, and further assume that and there is such that is Ahlfors -regular. Then
A simple modification of the proof of [23, Theorem 1], which is the Euclidean version of Theorem 1.1, shows that Theorem 4.1 is true when the assumption of Ahlfors regularity is replaced by the assumption that has the following covering property for sufficiently large values of the parameter :
Let . A subset of a metric space is -evenly coverable if there exists a constant such that for all sufficiently small , there exists a cover of by balls centered in such that
Hence, Theorem 4.1 follows from the following proposition.
Let be a metric space, and let be a bounded Ahlfors -regular subset of . Then is -evenly coverable for every .
Let . We assume that there is a constant such that for any and ,
In particular, this implies that .
Let , and consider a maximal -separated set in . Then covers , while is disjoint. Thus
This shows that satisfies conditions (i) and (ii) in the definition of -even coverability.
where is a subset of . To see that condition (iii) is verified, we must show that the cardinality of is bounded above by a number that does not depend on . Let . Then our assumption yields
This implies that
The desired bound on the cardinality of follows. ∎
We now turn to the question of sharpness in Theorem 1.1. We first prove Theorem 1.2 on the existence of mappings with upper gradients exhibiting optimal dimension increase. Note that the -Poincaré inequality is not assumed in Theorem 1.2.
Proof of Theorem 1.2.
The proof is a modified version of those appearing in  and . The main novelty is that we employ maximal separated sets in place of dyadic cubes; this in fact simplifies the proof. Let be a locally -Ahlfors regular metric measure space and let be a compact set with for some . We may assume without loss of generality that . By Frostman’s Lemma [25, Theorem 8.17], there is a finite and nontrivial Borel measure supported on such that
for each and .
For each , let be a maximal -separated set in ; we may assume that . Define
Note that each is finite. The local doubling condition on implies that there is a constant such that
for all and all .
For each , we may find a Lipschitz function such that , the support of is contained in , and
Here denotes the pointwise Lipschitz constant of the function , defined by
Let be a function. For each , define by
Now, define by
Then is continuous and bounded. Since is locally Lipschitz, the function is an upper gradient of . We claim that the sequence of norms is bounded. Using the bounded overlap condition (4.4), the Frostman condition (4.3), and Ahlfors -regularity, we calculate that
Our choice of implies that
and hence another application of the bounded overlap condition (4.4) shows that
These facts imply that is an upper gradient of .
We now choose the vectors randomly. More precisely, we assume that the functions are independent random variables distributed according to the uniform probability distribution on the closed unit ball , and hence the resulting function can also be considered as random variable; the expected value of this random variable is denoted by . We claim that almost surely. The desired result follows from this claim.
For denote by the -energy of a compactly supported Radon measure on a metric space , i.e.
If is finite, then the Hausdorff dimension of the support of is at least [25, Theorem 8.7].
We will prove that for every ,
which implies that almost surely; letting tend to will complete the proof.
By the Fubini–Tonelli theorem, it suffices to prove that
We denote by the maximum of the set of numbers Note that for some , since is finite. By [5, Lemma 4.4], it holds that
In view of this, it remains to show that
We will in fact show the stronger statement
Since this suffices.
Fix . For each , define by
Choose a ball that contains . Then , and so
For each , denote by the set of points for which . As above , where contains . Thus, by the above argument and the Frostman estimate (4.3),
Since , the final sum converges.
We now show that the set of Sobolev mappings that distort the Hausdorff dimension of a given set in the maximal way is prevalent, in the sense of Hunt–Sauer–Yorke ,  (see also , , and ).
To recall the notion of prevalence, let be a complete metric vector space (typically infinite dimensional). A compactly supported Borel measure on is said to be transverse to a Borel set if for every . A set is called to be shy if there exists a Borel set such that and a Borel measure that is transverse to . Using convolutions of measures it can be checked that the countable union of shy sets is again shy [22, 26]. A set is called prevalent if its complement is shy. Clearly, the countable intersections of prevalent sets is again prevalent and prevalent sets are dense in . If , then is prevalent if and only if it is a full Lebesgue measure set. The concept of prevalence has been introduced as a measure-theoretic notion of genericity in infinite dimensional spaces, especially function spaces. We will use this notion for the Newtonian–Sobolev space .
We consider a compact subset such that , and wish to show that the set of Newtonian Sobolev mappings with the property that is prevalent. Notice first that it is enough to show that
for each . Indeed, assuming that this is true we obtain prevalent subsets , , for which
Now set , which is again prevalent in as the countable intersection of prevalent sets. Letting in (4.7) we obtain that for all .
Note that by (4.5) there exists a continuous mapping with the property that
Here denotes the Frostman measure on , as in (4.3).
Statement (4.6) is implied by the following lemma.
Let satisfy (4.8). Denote by the set of all matrices with entries less than or equal to one in absolute value. Then for all the function satisfies
which in turn will follow from the boundedness of the triple integral
To prove (4.11) we will use the following
Let be a linear transformation from the set of matrices to and let be a fixed vector. Assume that the image of under contains a cube of width in . Then for we have
where is a constant depending only on and .
5. Regular foliations of a metric space
In this section we discuss bounds on dimension increase under Sobolev mappings for leaves in an -foliation of a metric space. In particular, we prove Theorem 1.4. Following the proof, we provide some comments regarding the limitations of that theorem and alternate methods to derive similar estimates.
The first step in the proof of Theorem 1.4 is the following lemma, which enables us to use Frostman’s lemma.
Assume the notation and hypotheses of Theorem 1.4. Then, for any compact set , the set
is a countable union of compact sets.
As is assumed to be proper, it suffices to show that is a countable union of closed sets. Since the -dimensional Hausdorff measure and the -dimensional Hausdorff content have the same null sets, it suffices to show that for each , the set
is closed. Let be a sequence converging to a point . Since and are continuous, for every , there is an index such that if , then
If , then there is a cover of by open balls such that
Since is compact, we may find such that the neighborhood is also covered by . This implies that
for all , which yields the desired contradiction. ∎
Proof of Theorem 1.4.
For ease of notation, denote
As we only consider the case that , it suffices to show that