Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates.
With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.
- 1 Introduction
- 2 Preliminaries
- 3 Weak Bakry-Émery curvature condition
- 4 BV class
- 5 Examples
In this paper we introduce and study functions of bounded variation on strongly local Dirichlet spaces which may not have Gaussian heat kernel bounds, but have sub-Gaussian heat kernel bounds as given in (3) and satisfy a weak Bakry-Émery curvature condition (2). We note that some properties that are usually taken for granted may not hold true in this setting because energy measures are not necessarily absolutely continuous with respect to a fixed measure . Therefore, unlike our analysis in  we can not develop and use locally Lipschitz functions, and need to develop a different set of tools.
This introduction is devoted to giving an overview of the content of the paper and to providing a summary of the main results obtained. The precise description of the sub-Gaussian heat kernel bounds and the definition of the Besov spaces that are investigated is presented in Section 2. Section 3 deals with a weak Bakry-Émery type curvature condition that is key in studying the notion of BV class introduced in Section 4. Eventually, Section 5 presents several examples of spaces where the theory developed in previous sections applies.
An approach to BV functions in metric measure spaces
Our approach to a theory of functions of bounded variation (BV) is based on the study of the Korevaar-Schoen class at the critical exponent. To explain our motivation and results, let us present this approach in the general context of metric measure spaces.
Let be a locally compact complete metric measure space where is a Radon measure. For and , we define the space as the collection of all functions for which
The –Korevaar-Schoen critical exponent of the space is then defined as
In the context of a complete metric measure space supporting a -Poincaré inequality and where is doubling, one has for every . Note that, at the critical exponent , one can construct a Dirichlet form
with domain by using a choice of a Cheeger differential structure as in . This Dirichlet form is then strictly local and the intrinsic distance associated to is bi-Lipschitz equivalent to the original metric . We refer to  and the references therein for further details. In that same context, at the critical exponent , one has and
By contrast, in the context of the present paper, is a complete metric measure space for which is doubling (even Ahlfors regular) but for which , where is a parameter called the walk dimension of the space, see [38, 40]. At the critical exponent one has a (strongly local but not strictly local) Dirichlet form
with domain whose heat kernel satisfies the sub-Gaussian estimates (3); see [44, Corollary 3.4] and Section 4, Remark 4.3. By analogy with the previous case, it seems then natural to study in that context the critical exponent and the associated class . For the spaces we are interested in, which are primarily fractals or products of fractals, we shall see that this class has many of the expected properties of a BV class: it has co-area formulas (see Theorem 4.15), existence of BV measures (see Theorem 4.23), and Sobolev embeddings (see Theorem 4.18). The key assumption that yields these properties is the weak Bakry-Émery condition (2).
A key point in the study of the Korevaar-Schoen classes is Proposition 4.1 which allows us to identify with the heat semigroup based Besov class , , that was introduced and extensively studied in our previous papers [1, 2]. In particular, we note that is therefore the critical parameter in the Besov scale of the classes , that is, for larger than this threshold the corresponding Besov classes contain only constant functions. Working directly with the Besov classes has the advantage that we are in the framework of , which allows us to use a wide range of heat semigroup techniques paralleling the methods developed in . For this reason most of our results are written for the Besov class and the corresponding critical exponent rather than in terms of and .
Weak Bakry-Émery nonnegative curvature condition
Our main tool in this paper is the heat semigroup. In the Euclidean case, the deep connection between regularizing properties of the heat semigroup and the theory of BV functions and sets of finite perimeter was uncovered by E. De Giorgi in the celebrated paper . Among many other works, this connection was further developed and investigated by M. Ledoux in  (see also the references therein). The Bakry-Émery calculus shows that regularizing properties of the heat semigroup are intimately connected with Ricci curvature-type lower bounds on the underlying space, see . Thus, it should come as no surprise that the approaches of De Giorgi and Ledoux and the notions of isoperimetric inequalities and BV functions may be generalized to large classes of spaces for which Ricci curvature type lower bounds are well understood, like the now-extensively studied RCD spaces (see [7, 28]) or sub-Riemannian spaces (see [25, 23]). In the context of the present paper, although the Bakry-Émery calculus is not available, the weak Bakry-Émery curvature condition introduced in  has a natural Hölder analogue which is the key assumption of our work. We shall say that the weak Bakry-Émery non-negative curvature condition is satisfied if there exist a constant and a parameter such that for every , and ,
We prove in Theorem 3.7 that fractional metric spaces for which must satisfy with . Note that the hypothesis ofTheorem 3.7 is stable under rough isometries in the sense of Barlow-Bass-Kumagai [22, 18, 17] and is a part of the general theory of fractional diffusions [20, 13, 12, 38, 50, 51, 10, 15, 16, 19, and references therein]. For nested fractals, Theorem 3.7 yields that is satisfied with . Moreover, in that case the value is optimal in the sense that is not satisfied for . Theorem 3.7 also proves that the Sierpinski carpet satisfies with , however we conjecture that in fact the Sierpinski carpet satisfies with . It will be a subject of future work to investigate whether .
We would like to briefly comment on the curvature interpretation of the weak Bakry-Émery condition . Our Theorem 3.7 holds only when , which means that this theorem is applicable for low dimensional spaces. In such a low dimensional situation, geometrically speaking, there is no curvature. For nested fractals, the topological and topological Hausdorff dimension are both 1. Thus, in some sense, they are analogues of lines which have zero curvature. From a different perspective this corresponds to the Hodge-type theorems in [42, 43] and Liouville-type theorems [36, 45] and [67, Introduction and Section 4]. The curvature interpretation of will only manifest itself in higher dimension when , and this is why Subsection 3.3 is important, as it allows us to construct higher dimensional examples satisfying . When one can expect the boundary of sets of finite perimeter to have a real geometry more complicated than that of Cantor sets.
Beyond direct products of fractional spaces [46, 70, 27, 69], which in a sense are still flat, one could try to construct “fractional manifolds, or fractafolds [68, 71], with non-negative curvature” which would be metric spaces with heat kernels satisfying a sub-Gaussian estimate and a geometric non-negative curvature condition, which would in turn imply the validity of on these fractafolds. This is beyond the scope of the present paper.
One long term goal of this project is also to develop tools for Li-Yau type estimates, in particular on the decay of the gradient of the heat kernel. In our setting a gradient is to be understood in a measure-theoretic sense, which motivates a large part of our work. Consideration of fractals in this context is important, in particular, because they appear as models for manifolds with slow heat kernel decay, , and limit sets of Schreier graphs of self-similar groups, which include groups of intermediate growth and non elementary amenable groups, [61, 60, 21, 48, and references therein]. The forthcoming papers [4, 5] will extend these ideas to non-local forms and infinite dimensional spaces.
Main results on BV functions under the condition
On a -Ahlfors regular metric measure space whose heat kernel satisfies the sub-Gaussian estimates (3), we define
and for ,
A set will be said to be of finite perimeter if . For a set of finite perimeter we define its perimeter as . Note that, unlike in the strictly local setting in , the perimeter may not be induced by a Radon measure in a classical sense, but in some generalized sense that will be further studied in some specific situations in .
Our main assumption to study the BV class is that satisfies with
Locality property (Theorem 4.9) There is a constant such that for every ,
Co-area estimate (Theorem 4.15) There exist constants such that for every non-negative ,
where . In particular, for the sets are of finite perimeter for almost every .
(Theorem 4.17) There exists a constant such that for every Borel set ,
where denotes the -codimensional lower Minskowski content of . In particular, any set whose measure-theoretic boundary has finite -codimensional lower Minskowski content has finite perimeter
Sobolev inequality I (Theorem 4.18) Assume . Then and there is such that for every ,
where the critical Sobolev exponent is given by the formula
In particular, there exists a constant such that for every set of finite perimeter,
This is our analog of an isoperimetric inequality.
Sobolev inequality II (Theorem 4.19) Assume . Then and there exists a constant such that for every , and a.e.
Note that this is in contrast to the strictly local case, where no such pointwise control can be obtained for BV functions; however, in that case, if also supports a -Poincaré inequality, then we have a pointwise control in terms of the Hardy-Littlewood maximal function of the BV energy measure.
We also show that BV functions naturally induce Radon measures on that we call BV measures, see Section 4.6. In a certain sense, those measures can be thought of as gradient measures of BV functions. Because of possible oscillatory phenomena due to the geometry of the underlying space , we do not expect that a given has in general a unique associated BV measure. However, Theorem 4.23 shows the remarkable fact that all the BV measures associated to a given are mutually equivalent. If the function is regular enough we show in Theorem 4.29 that its energy measure can be controlled by the lower envelope of its BV measures.
The motivation for this paper comes from the following three standard fractal examples: unbounded Vicsek set (Figure 1), unbounded Sierpinski gasket (Figure 2), and unbounded Sierpinski carpet (Figure 3). The properties of BV spaces are remarkably different in these cases and, therefore, on spaces with sub-Gaussian heat kernel bounds (3) one can expect a theory of BV functions that in general is analogous to the BV theory in , but in some sense richer in detail and more variable.
On the Vicsek set, as on all nested fractals, and BV functions are dense in , see Theorems 5.1 and 5.2(1), with equivalent BV and energy measures. Furthermore, in a future work on the Vicsek set one will see that we can develop a complete theory analogous to the one dimensional case but including new oscillatory phenomena.
On the Sierpinski gasket we also have , but BV functions do not even contain piecewise harmonic functions, see Theorems 5.1 and 5.2(2), and one expects all BV functions to be discontinuous, and BV measures to be purely atomic, see Conjecture 5.3. The absence of intrinsically smooth functions of bounded variation is a very surprising phenomenon that has not been observed before. On the Sierpinski gasket the proof of this part of the theorem relies on delicate results following from the Furstenberg-Kesten theory of invariant measures and Lyapunov exponents for products of i.i.d. random matrices, which includes a non-commutative matrix version of the classical ergodic theorems. In particular, one can expect that the estimates of the Lyapunov exponents are intimately related to the Besov-type estimates of the intrinsically smooth functions. This will be the subject of future work in .
On the Sierpinski carpet we conjecture that , c.f. Conjecture 5.4. Proving this fact would involve, in particular, improving estimates on the Hölder continuity of harmonic functions obtained in [15, 12, 16]. It is also closely related to the Besov critical exponents defined in (13), and to a measurable version of isoperimetric type arguments that will be the subject of future work. In addition, this conjecture combines the fractal analog of the Einstein-type relation between spatial distance and time with a detailed analysis of the -neighborhoods of boundaries of open sets and our co-area formulas in Section 4.3.
Convention and notations
Throughout the paper we use to denote positive constants that may change from line to line. The quadruple denotes a topological measure space with a Dirichlet form on , with the collection of functions for which is finite. The notations and denote the Hausdorff, walk, and topological-Hausdorff dimensions respectively, of , when is equipped with a metric ; so strictly speaking, we deal with a metric measure space equipped with a Dirichlet form . We will assume throughout that is associated with the metric via the sub-Gaussian estimates (3) and the weak Bakry-Émery curvature conditions (2) mentioned above.
The authors thank Prof. Martin Barlow for helpful information and Prof. Naotaka Kajino for many stimulating and helpful discussions.
P.A-R. was partly supported by the Feodor Lynen Fellowship, Alexander von Humboldt Foundation (Germany). F.B. was partly supported by the grant DMS #1660031 of the NSF (U.S.A.) and a Simons Foundation Collaboration grant. L.R. was partly supported by the grant DMS #1659643 of the NSF (U.S.A.). N.S. was partly supported by the grant DMS #1800161 of the NSF (U.S.A.). A.T. was partly supported by the grant DMS #1613025 of the NSF (U.S.A.).
Our assumptions are quite general, and the main classes of examples we are interested in this paper are fractal spaces. We refer to [12, 38, 50, 51] for further details on the following framework and assumptions.
2.1 Metric measure Dirichlet spaces with sub-Gaussian heat kernel estimates
Let be a locally compact metric measure space where is a Radon measure supported on . Let now be a Dirichlet form on , that is: a densely defined, closed, symmetric and Markovian form on , see [35, 30]. We denote by the vector space of all continuous functions with compact support in and its closure with respect to the supremum norm. A core for is a subset of which is dense in in the supremum norm and dense in in the norm
In the literature, this norm is sometimes denoted or . The Dirichlet form is called regular if it admits a core. It is called strongly local if for any two functions with compact supports such that is constant in a neighborhood of the support of , we have , see [35, Page 6]. We denote by the heat semigroup associated with the Dirichlet space and refer to Section 2.2 in  for a summary of its basic properties.
Throughout the paper, we make the following assumptions. Note that they are not independent (some may be derived from combinations of the others); the list was chosen for comprehension rather than minimality.
Assumption 2.1 (Regularity).
has compact closure for any and any ;
is Ahlfors -regular, i.e. there exist such that for any .
is a regular, strongly local Dirichlet form.
Assumption 2.2 (Sub-Gaussian Heat Kernel Estimates).
has a continuous heat kernel satisfying, for some and ,
for -a.e. and each .
The parameter is the Hausdorff dimension, and the parameter is called the walk dimension, even though it is, strictly speaking, not a dimension of a geometric object. It is possible to prove that if the metric space satisfies a chain condition, then , see [37, 14, 39]. When , one speaks of Gaussian estimates and when , one speaks then of sub-Gaussian estimates. In this framework, it is known that the semigroup is conservative, i.e. . It is important to note that in this paper, unlike , the distance is not necessarily the intrinsic distance associated with the Dirichlet form. The only link between the distance and the Dirichlet form we need to develop our theory is the sub-Gaussian heat kernel estimates (3).
The following is an easy but frequently-used consequence of the sub-Gaussian bounds for the heat kernel.
Let . There are constants so that for every , and
Using the sub-Gaussian estimate (3) to bound from above and from below, this reduces to the observation that
for a suitable choice of and . ∎
2.2 Barlow’s fractional metric spaces and fractional diffusions
Our setting is closely related to the setting of fractional metric spaces with fractional diffusions defined by Barlow , which differ from ours only in that they are assumed to be geodesic; later we will see that this latter assumption implies, and is perhaps no stronger than, the weak Bakry-Émery assumption we need in order to construct a rich theory of BV functions.
According to [12, Definition 3.2], a complete metric measure space with a Borel measure is a fractional metric space of dimension if it is -Ahlfors regular and satisfies the midpoint property, i.e. for any there exists such that . The latter is equivalent to requiring the space be geodesic. In this context, Barlow introduced in [12, Section 3] a class of processes called fractional diffusions. A fractional diffusion is a -symmetric, conservative Feller diffusion on for which there is a jointly continuous heat kernel that is symmetric, has the semigroup property and satisfies the sub-Gaussian estimates (3). In his notation, our is in provided is geodesic. We note that in this context [12, Theorem 3.20] says .
2.3 Heat kernel based Besov classes
Let and . As in , we define the Besov seminorm
and define the heat semigroup-based Besov class by
Our first goal is to compare the space to Besov type spaces previously considered in a similar framework (see ).
For and , we introduce the following seminorm: for ,
for , and
We then define the Besov space by
It is clear that for with .
[62, Theorem 3.2] Let and . We have and there exist constants such that for every and ,
In particular, , and is in if and only if
The above theorem is essentially a rephrasing of [62, Theorem 3.2], though the estimates we obtain are slightly sharper. However, the notion of Besov spaces given in  considers dyadic jumps in the parameter ; hence the proof given there is slightly more complicated than ours. We include the relatively short proof because the estimates are useful in later sections.
We first prove the lower bound. For and observe from (3) that
from which the lower bound follows. We now turn to the upper bound. Fixing , we set
so that . By (3) and the inequality ,
where and .
On the other hand, for , by (3) we have
Hence one has
The proof is thus complete. ∎
2.4 Besov regularity of indicators of sets and density of in
In order to have an interesting theory we certainly need our Besov spaces to contain non-constant functions. It is also natural to be concerned with whether they are dense in the Lebesgue spaces. Accordingly we follow [1, Section 5.2] and define critical exponents as follows.
Note that . In this section we concern ourselves only with a simple condition for to be dense in , as this is the essential case to consider in studying functions of bounded variation. It will become apparent when we treat the co-area formula, Theorem 4.15 that the significant question is whether a characteristic function of a Borel set is in , and it is well-known that this is related to boundary regularity of the measure-theoretic boundary.
Let be a Borel set. We say is a density point of and write if
The measure-theoretic boundary is , see e.g. [6, Section 4]. Now for define the measure-theoretic -neighborhood by
where and similarly for .
Notice that , where this last is the topological boundary. We have the following easy consequence of Theorem 2.4.
Suppose is a finite measure Borel set such that