Poincaré-type inequalities and finding good parameterizations
A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. In 1960, E. R. Reifenberg proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-Hölder image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an -Ahlfors regular rectifiable set that satisfies a Poincaré-type inequality involving the tangential derivative. Then, we show that a Carleson-type condition on the oscillations of the tangent planes of guarantees that is contained in a bi-Lipschitz image of an -plane. We also explore the Poincaré-type inequality considered here and show that it is in fact equivalent to other Poincaré-type inequalities considered on general metric measure spaces.
Key words and phrases:Rectifiable set, Carleson-type condition, Poincaré-type condition, -Poincaré inequality, -Poincaré inequality, Ahlfors regular, bi-Lipschitz image
Finding bi-Lipschitz parameterizations of sets is a central question in areas of geometric measure theory and geometric analysis. A Lipschitz function on a metric space plays the role played by a smooth function on a manifold, and a bi-Lipschitz function plays the role of that of a diffeomorphism. Many concepts in metric spaces, such as metric dimensions and Poincaré inequalities, are preserved under bi-Lipschitz mappings. Moreover, a bi-Lipschitz parameterization of a set by Euclidean space leads to its uniform rectifiability. Uniform rectifiability is a quantified version of rectifiability which is well adapted to the study of problems in harmonic analysis on non-smooth sets.
The type of parameterizations discussed in this paper first appeared in 1960 when Reifenberg [Rei60] showed that if a closed set is well approximated by affine -planes at every point and every scale, then is a bi-Hölder image of . Such a set is called a Reifenberg flat set. In recent years, there has been renewed interest in this result and its proof. In particular, Reifenberg type parameterizations have been used to get good parameterizations of many spaces such as chord arc surfaces with small constant (see [Sem91a, Sem91b]), and limits of manifolds with Ricci curvature bounded from below (see [CC97, CN13]). Moreover, Reifenberg’s theorem has been refined to get better parameterizations of a set: bi-Lipschitz parameterizations (see [DS91], [Tor97], [DT12], [Mer15]). In fact, it is well known today, due to the authors of the latter references, that Carleson-type conditions are the correct conditions to study when seeking necessary and sufficient conditions for bi-Lipschitz parameterizations of sets. For example, in [Tor97], Toro considers a Carleson condition on the Reifenberg flatness of that guarantees its bi-Lipschitz parameterization. In [DT12], David and Toro consider a Carleson condition on the Jones beta numbers and on the (possibly smaller) numbers that guarantees the same result. In [Mer15], the author studies a Carleson-type condition on the oscillation of the unit normals to an -rectifiable set of co-dimension 1, that guarantees its bi-Lipschitz parameterization. An -rectifiable set is a generalization of a smooth -manifold in . Rectifiable sets are characterized by having (approximate) tangent planes (see Definition 2.4) at -almost every point. Moreover, in the special case when the rectifiable set has co-dimension 1, then has an (approximate) unit normal (see Remark 2.5) at -almost every point. In fact, in [Mer15], the author considers an -Ahlfors regular rectifiable set , of co-dimension 1, that satisfies the following Poincaré-type inequality for and :
For all , , and a Lipschitz function on , we have
where denotes the Poincaré constant that appears here, is the dilation constant, is the Hausdorff measure restricted to , is the average of the function on , is the Euclidean ball in the ambient space , and denotes the tangential derivative of (see Definition 2.6).
Then, the author shows that a Carleson-type condition on the oscillation of the unit normal to guarantee a bi-Lipschitz parameterization of .
(see [Mer15], Theorem 1.5) Let be an -Ahlfors regular rectifiable set containing the origin, and let be the Hausdorff measure restricted to . Assume that satisfies the Poincaré-type inequality (1.1) with and . There exists , such that if for some choice of unit normal to , we have
then is contained in the image of an affine -plane by a bi-Lipschitz mapping, with bi-Lipschitz constant depending only on , and .
In this paper, we generalize Theorem 1.1 to higher co-dimensions and arbitrary dilation constants . Before stating the theorem, let us introduce some notation. Suppose that is an -Ahlfors regular rectifiable set that satisfies the Poincaré-type inequality (1.1). Fix and . Let such that the approximate tangent plane of at the point exists, and denote by the orthogonal projection of on . Using the standard basis of , , we can view as an matrix whose column is the vector is . Thus, we denote by the matrix . Finally, let , be the matrix whose entry is the average of the function in the ball .
Let be an -Ahlfors regular rectifiable set containing the origin, and let be the Hausdorff measure restricted to . Assume that satisfies the Poincaré-type inequality (1.1). There exist and , such that if
where denotes the Frobenius norm 111 of , then there exists an onto -bi-Lipschitz map where the bi-Lipschitz constant and an -dimensional plane , with the following properties:
where . Moreover,
Notice that the conclusion of Theorem 1.2 states that is (locally) contained in a bi-Lipschitz image of an -plane instead of being exactly a (local) bi-Lipschitz image of an -plane. This is very much expected, since we do not assume that is Reifenberg flat, and thus we have to deal with the fact that might have holes. However, if we assume, in addition to the hypothesis of Theorem 1.2, that is Reifenberg flat, then we do obtain that is in fact (locally) a bi-Lipschitz image of an -plane. We show this in this paper as a corollary to Theorem 1.2.
A natural question is whether the hypotheses of Theorem 1.2, that is the Ahlfors regularity of , the Poincaré inequality (1.1), and the Carleson condition (1.3) imply that is Reifenberg flat. An affirmative answer to this question would directly imply (by the paragraph above) that the conclusion of Theorem 1.2 should be that is exactly a bi-Lipschitz image of an -plane instead of being just contained in bi-Lipschitz image of an -plane. A negative answer would show that the conclusion of Theorem 1.2 is the best that we can hope for. It is not surprising that the Poincaré inequality (1.1) is the correct condition to explore in order to answer this question (which as we discuss below, will turn out negative). In fact, it is already known that (1.1) encodes geometric properties of the set .
Let be a metric measure space, where is an -Ahlfors regular rectifiable set in , is the measure that lives on , and is the metric on which is the restriction of the standard Euclidean metric on . In [Mer15], the author proves that the Poincaré inequality (1.1) implies that is quasiconvex. More precisely,
A metric space is -quasiconvex if there exists a constant such that for any two points and in , there exists a rectifiable curve in , joining and , such that .
(see [Mer15] Theorem 5.5) 222 Notice that Theorem 5.5 in [Mer15] is stated and proved in the ambient space (so ) and for . However, the proof of Theorem 5.5 in [Mer15] is independent from the co-dimension of . Thus the exact same statement holds here in the higher co-dimension case, and the quasiconvexity constant stays independent of . Moreover, it is very easy to see that Theorem 5.5 in [Mer15] still holds with arbitrary , and in that case, would also depend on . Let be as discussed above. Suppose that satisfies the Poincaré-type inequality (1.1). Then is -quasiconvex, with .
There are many Poincaré-type inequalities found in literature that imply quasiconvexity (see for example [Che99], [DCJS13], [Kei03], [Kei04]). To state a couple of the main ones, let be a measure space endowed with a metric and a positive complete Borel regular measure supported on . Denote by the metric ball in , center and radius . Moreover, assume that for all and .
Let . is said to admit a -Poincaré inequality if there exist constants and such that for any measurable function and for any upper gradient (see Definition 2.12) of , the following holds
where , , and .
Let . is said to admit a -Poincaré inequality if there exist constants and such that for every Lipschitz function on , and for every and , we have
(see Definition 2.15 for the definition of ).
These Poincaré inequalities are a-priori different because the right hand side varies according to the notion of “derivative” used on the metric space. However, Keith has shown (see [Kei03], [Kei04]) that if is a complete metric measure space with a doubling measure, then and are equivalent. It turns out that the Poincaré-type inequality (1.1) is also related to (1.8) and (1.9).
In this paper, we take as described above and prove that in this setting, the Poincaré-type inequalities (1.1) (or a more generalized version of it, see (1.12) below), (1.8), and (1.9) are equivalent.
Let , and let be a metric measure space, where is an -Ahlfors regular rectifiable set in , is the measure that lives on , and is the metric on which is the restriction of the standard Euclidean metric on .Then, the following are equivalent:
There exist constants and such that for any measurable function , for any upper gradient of , and for every and , we have
There exist constants , and , such that for every Lipschitz function on , and for every and , we have
There exist constants , and , such that for every Lipschitz function on , and for every and , we have
Theorem 1.7 is interesting in its own right, as it shows that the Poincaré inequality (1.1) (or more generally, (1.12) ) is equivalent to the other usual Poincaré-type inequalities on metric spaces that imply quasiconvexity. Moreover, Theorem 1.7 opens the door to many examples of spaces satisfying the Poincaré inequality (1.12) as there are many examples in literature of spaces satisfying the -Poincaré and -Poincaré inequalities (see for example [BS07], [HK00], [BB11], [Laa00]). This allows us to get an example of a set that is not Reifenberg flat, and yet satisfies all the hypotheses of Theorem 1.2.
There exists a non-Reifenberg flat, -Ahlfors regular, rectifiable set that satisfies all the hypotheses of Theorem 1.2.
The paper is structured as follows: in Section 2, we introduce some definitions and preliminaries. In Section 3, we prove Theorem 1.2. Moreover, we prove that Theorem 1.1 follows as a corollary from Theorem 1.2. Section 4 is dedicated to proving that the Poincaré inequality (1.12) is equivalent to the -Poincaré and the -Poincaré inequalities. Finally, in the last section, we prove Theorem 1.8 by constructing a concrete example of a set that is not Reifenberg flat, yet satisfies the hypotheses of Theorem 1.2.
Throughout this paper, our ambient space is . denotes the open ball center and radius in , while denotes the closed ball center and radius in . denotes the distance function from a point to a set. is the -Hausdorff measure. Finally, constants may vary from line to line, and the parameters they depend on will always be specified in a bracket. For example, will be a constant that depends on and that may vary from line to line.
We begin by the definitions needed starting section 3 and onwards.
Let . A function is called Lipschitz if there exists a constant , such that for all we have
The smallest such constant is called the Lipschitz constant and is denoted by .
A function is called -bi-Lipschitz if there exists a constant , such that for all we have
Let’s introduce the class of n-rectifiable sets, and the definition of approximate tangent planes.
Let be an -measurable set. is said to be countably n-rectifiable if
where , and is Lipschitz, and , for
If is an -measurable subset of . We say that the -dimensional subspace is the approximate tangent space of at , if
Notice that if it exists, is unique. From now on, we shall denote the tangent space of at by . Moreover, in the special case when has co-dimension 1, then one can define the unit normal to at the point to be the unit normal to . Thus, the unit normal exists at every point that admits a tangent plane, and of course, there are two choices for the direction of the unit normal.
It is well known (see [Sim83]; Theorem 11.6) that -rectifiable sets have tangent planes at almost every point in the set.
Let be a real valued Lipschitz function on . The tangential derivative of at the point id denoted by and defined as follows:
where , is the restriction of on the affine subspace , and is the usual gradient of .
In the special case when is a smooth function on , we have
where is the orthogonal projection of on , and is the usual gradient of .
Note that exists at - almost every point in .
We also need to define the notion of Reifenberg flatness:
Let be an -dimensional subset of . We say that is -Reifenberg flat for some , if for every and , we can find an -dimensional affine subspace of that contains such that
Notice that the above definition is only interesting if is small, since any set is 1-Reifenberg flat.
In the proof of our Theorem 1.2, we need to measure the distance between two -dimensional planes. We do so in terms of normalized local Hausdorff distance:
Let be a point in and let . Consider two closed sets such that both sets meet the ball . Then,
is called the normalized Hausdorff distance between and in .
Let us recall the definition of an -Ahlfors regular measure and an -Ahlfors regular set:
Let be a closed, measurable set, and let be the -Hausdorff measure restricted to . We say that is -Ahlfors regular if there exists a constant , such that for every and , we have
In such a case, the set is called an -Ahlfors regular set, and is referred to as the Ahlfors regularity constant.
Let us now move to definitions and notations needed in sections 4 and 5. In these sections, denotes a space endowed with a metric . denotes the open metric ball of center and radius . Moreover, denotes a measure space endowed with a metric and a positive complete Borel regular measure supported on such that for all and .
Let be a metric measure space. We say that is a doubling measure if there is a constant such that
where , .
In sections 4 and 5, a curve in a metric space is a continuous non-constant map from a compact interval into . is said to be rectifiable if it has finite length, where the latter is denoted by . Thus, any rectifiable curve can be parametrized by arc length, and we will always assume that it is.
Let us now define the notions of upper gradients, -weak upper gradients, and the Local Lipschitz constant function.
A non-negative Borel function is said to be an upper gradient of a function if
for any rectifiable curve .
Let and let be a family of rectifiable curves on . We define the -modulus of by
where the infimum is taken over all nonnegative Borel functions such that for all .
A non-negative measurable function is said to be a p-weak upper gradient of a function if
for -a.e. rectifiable curve (that is, with the exception of a curve family of zero -modulus).
Let be a Lipschitz function on a metric measure space . The local Lipschitz constant function of is defined as follows
where denotes the metric ball in , center , and radius .
Let us note here that for any Lipschitz function , denotes the usual Lipschitz constant (see sentence below (2.1)), whereas stands for the local Lipschitz constant function defined above.
3. A bi-Lipschitz parameterization of
The main goal in this section is to prove Theorem 1.2. We begin with three linear Algebra lemmas needed to prove the theorem, as they can be stated and proved independently.
In the next lemma, let be an -dimensional subspace of . Denote by the orthogonal projection on . Then, there exists a , such that for any , and for any linear operator on such that
where denotes the induced operator norm, has exactly eigenvalues such that
and exactly eigenvalues , such that
Since is an orthogonal projection, then there exists an orthonormal basis of such that the matrix representation of in this basis is
where denotes the identity matrix.
Let (with to be determined later), and suppose is as in the statement of the lemma. Let be the matrix representation of in the basis . Then, by (3.1), we have
Choosing such that , we can guarantee that is disjoint from . Thus, by the Gershgorin circle theorem (see [LeV07], p.277-278), contains exactly eigenvalues of , and contains exactly eigenvalues of . The lemma follows from (3.6), (3.7) and (3.8) ∎
Let be an affine subspace of of dimension , . Denote by , the -neighborhood of , that is,
(see [Mer15], Lemma 3.1) 333Notice that Lemma 3.1 in [Mer15] is stated and proved in the ambient space , whereas Lemma 3.2 here has as the ambient space. However, one can very easily adapt the same proof of Lemma 3.1 in [Mer15] to this higher co-dimension case here, while noticing that in the latter case should also depend on the co-dimension . Let be an -Ahlfors regular subset of , and let be the Hausdorff measure restricted to . There exists a constant such that the following is true: Fix , and let . Then, for every , an affine subspace of of dimension , there exists such that and .
(see [Mer15] Lemma 3.3) 444Notice that Lemma 3.3 in [Mer15] is stated and proved in the ambient space , whereas Lemma 3.3 here has as the ambient space. However, the proof of Lemma 3.3 in [Mer15] is in fact independent from the co-dimension of . Thus the exact same proof holds here, and the constant stays independent of . Fix , and let be vectors in . Suppose there exists a constant such that
Moreover, suppose there exists a constant , such that
Then, for every vector , can be written uniquely as
with being a constant depending only on , , and .
Throughout the rest of the paper, denotes an -Ahlfors regular rectifiable subset of and denotes the Hausdorff measure restricted to . The average of a function on the ball is denoted by
We recall the statement of Theorem 1.2: if satisfies the Poincaré-type condition (1.1), and if the Carleson-type condition (1.3) on the oscillation of the tangent planes to is satisfied, and if then is contained in a bi-Lipschitz image of an -dimensional plane.
To prove this theorem, we follow steps similar to those used in [Mer15] to prove the co-dimension 1 case (see Theorem 1.5 in [Mer15]) which is stated as Theorem 1.1 in this paper. First, we define what we call the -numbers
where , and , has as its matrix representation in the standard basis of , and is the matrix whose entry is the average of the function in the ball .
These numbers are the key ingredient to proving our theorem. In Lemma 3.4, we show that the Carleson condition (1.3) implies that these numbers are small at every point and every scale . Moreover, for every point , and series is finite. Then, in Theorem 3.5, we use the Poincaré-type inequality to get an -plane at every point and every scale such that the distance (in integral form) from to is bounded by . This means, by Lemma 3.4, that those distances are small, and for a fixed point , when we add these distances at the scales for , this series is finite 555 A note for the interested reader: Theorem 3.5 implies that the series is finite. See [Mer15] on how this relates to the -numbers, and the theorems found in [DT12] that involve a Carleson condition on the -numbers that guarantees a bi-Lipschitz parameterization of the set.. Theorem 3.5 is the key point that allows us to use the bi-Lipschitz parameterization that G. David and T. Toro construct in [DT12]. In fact, what they do is construct approximating -planes, and prove that at any two points that are close together, the two planes associated to these points at the same scale, or at two consecutive scales are close in the Hausdorff distance sense. From there, they construct a bi-Hölder parameterization for . Then, they show that the sum of these distances at scales for is finite (uniformly for every ). This is what is needed for their parameterization to be bi-Lipschitz (see Theorem 3.7 below and the definition before it). Thus, the rest of the proof is devoted to using Theorem 3.5 in order to prove the compatibility conditions between the approximating planes mentioned above.
Note that, in the process of proving Theorem 1.2, we find several parts of the proof very similar to the proof of the co-dimension 1 case found in [Mer15] (see Theorem 1.5 in [Mer15] or Theorem 1.1 in this paper). In fact, most of the differences in the proof happen in Lemma 3.4 and Theorem 3.5, with the most important difference being in the latter. The rest of the proof follows closely to the proof of co-dimension 1 case. Thus, in this paper we do as follows: first, we prove Lemma 3.4 and Theorem 3.5 and include all the details. Then, for the rest of the proof (that is introducing the David and Toro bi-Lipschitz construction, and proving the compatibility conditions between the approximating planes that allow us to use this construction), we only give an outline of the main ideas, and leave the smaller details and tedious calculations out. However, in each place where the details are omitted, we refer the reader to the parts of the proof of Theorem 1.5 in [Mer15] where they can be found. That being said, this part of the proof of Theorem 1.2 still has enough details so that the reader understands all the steps needed to get the bi-Lipschitz parameterization of , and the intuition behind them. Moreover, the way the proof is presented here includes all the information that we need from the construction of the bi-Lipschitz parameterization of to prove the corollaries that follow from Theorem 1.2.
Let be an -Ahlfors regular rectifiable set containing the origin, and let be the Hausdorff measure restricted to . Let , and suppose that
Then, for every , we have
where the -numbers are as defined in (3.15) and . Moreover, for every and , we have
where and .
Fix , and fix . For all , and for all , we have
This is a straightforward computation that uses (3.20) and the Ahlfors regularity of , and is found in details in [Mer15] (see [Mer15], Lemma 4.1 proof of inequality (4.6)). Moreover, it is trivial to check that
which is exactly (3.17).
To prove inequality (3.18), fix and