The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces

Xiangyun Xie; Yu Liu; Pengtao Li; Jizheng Huang

doi:10.1515/anona-2023-0119

Article Open Access

The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces

Xiangyun Xie , Yu Liu , Pengtao Li and Jizheng Huang

Published/Copyright: February 5, 2024

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From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak ( 1 , 2 ) -Poincaré inequality instead of the weak ( 1 , 1 ) -Poincaré inequality compared with the results in the previous references.

Keywords: BV space; Dirichlet space; heat kernel; capacity; Sobolev inequality

MSC 2010: 28A12; 31E05; 26B30

1 Introduction

Functions of bounded variation (bounded variation [BV] functions) play an important role in several problems of calculus of variations, for example, minimal area problems and free discontinuity problems due to their connection to finite perimeters and rectifiable sets. Many references focus on the study of BV functions in the Euclidean setting. The extension of the Euclidean BV theory to metric spaces has become a recent interesting topics (see [7,8,39] for theories of BV functions on metric measure spaces). Furthermore, as an important concept of geometric measure theory on metric spaces, it is natural to consider the BV capacity as the continuation of the research of BV functions. The BV capacity on metric spaces has been investigated in [24,25,28,34] and the references therein. It should be noted that the typical assumptions on the metric space proposed in their references are that support the weak ( 1 , 1 ) -Poincaré inequality. Recently, a series of articles consider hot topics for Besov spaces, Sobolev spaces, and BV spaces on the Dirichlet space under some extra assumptions, which is a typical example of metric spaces (cf. [1–6]). They obtain some of their results on the Dirichlet space satisfying the weak ( 1 , 2 ) -Poincaré inequality instead of the weak ( 1 , 1 ) -Poincaré inequality, where the former is weaker than the latter.

In the study of partial differential equations and potential theory on Euclidean spaces, the Sobolev embedding inequality is usually used as an important tool. As an application of the Sobolev embedding inequality, we immediately deduce the isoperimetric inequality for the sets of finite perimeter, which is one of essential geometric results. In recent decades, the Sobolev embedding inequality and the isoperimetric inequality have been generalized to metric measure spaces. Let ( X , μ , ℰ , ℱ ) be a complete metric measure Dirichlet space that satisfies a weak Bakry-Émery estimate and the volume growth condition μ ( B ( x , r ) ) ≥ C r Q with r > 0 for some Q > 0 . [2, Theorem 5.3], together with [12, Theorem 3.6], implies the following Sobolev inequality:

(1) ‖ f ‖ Q ∕ ( Q − 1 ) ≤ C ‖ D f ‖ ( X ) , f ∈ BV ( X ) ,

and the isoperimetric inequality: for any Borel set E ⊂ X with finite perimeter is

(2) [ μ ( E ) ] ( Q − 1 ) ∕ Q ≤ C ℳ − ( E ) ,

where ‖ D f ‖ ( X ) is the total variation of f and ℳ − ( E ) denotes the upper Minkowski content of E (see Proposition 4.2).

Based on the aforementioned results obtained in the study by Alonso-Ruiz et al. [2], the goal of this article is to establish further characterizations of inequalities (1) and (2) by the aid of the BV capacity on the Dirichlet space ( X , μ , ℰ , ℱ ) , where X is a locally compact metric space equipped with a doubling Radon measure μ and a strictly local Dirichlet form ℰ on L 2 ( X ) . Denote by cap ( E , BV ( X ) ) the BV capacity for arbitrary set E ⊂ X . Under the additional assumption of the weak Bakry-Émery estimate, we split equality (1) to obtain some analytic inequalities and geometric inequalities in Theorems 4.3 and 4.4. Roughly speaking, the capacity inequalities in Theorems 4.3 and 4.4 provide a description of the gap between L Q ∕ ( Q − 1 ) ( X ) and BV ( X ) . Recently, Liu et al. [37] obtained the capacitary inequalities for the variational p -capacity on the complete doubling metric measure space supporting the weak ( 1 , p ) -Poincaré inequality with p ∈ [ 1 , ∞ ) . These results generalize the works of Xiao and Liu obtained in [43,44] on the Euclidean space and in [36] on the Grushin plane, respectively. Furthermore, we also obtain the Maz’ya-Sobolev inequalities on the Dirichlet space under same conditions.

In what follows, we recall some facts and assumptions for Dirichlet spaces introduced in [1–6] in order to state our results.

Let X be a locally compact metric space equipped with a Radon measure μ supported on X . Let ( ℰ , ℱ = dom ( ℰ ) ) be a Dirichlet form on X , meaning it is a densely defined, closed, symmetric, and Markovian form on L 2 ( X ) .

Denote by C c ( X ) the space of all continuous functions with compact support in X and by C 0 ( X ) its closure with respect to the supremum norm. A core for ( X , μ , ℰ , ℱ ) is a subset C of C c ( X ) ∩ ℱ , which is dense in C c ( X ) in the supremum norm and dense in ℱ in the norm

( ‖ f ‖ L 2 ( X ) 2 + ℰ ( f , f ) ) 1 ∕ 2 .

The Dirichlet form ℰ is called regular if it admits a core. It is called strongly local if for any two functions u , v ∈ ℱ with compact supports such that u is constant in a neighborhood of the support of v , we have ℰ ( u , v ) = 0 .

Throughout this article, we assume that ( ℰ , ℱ ) is a strongly local regular Dirichlet form on L 2 ( X ) . Since ℰ is regular, for every u , v ∈ ℱ ∩ L ∞ ( X ) , the energy measure Γ ( u , v ) is defined via the formula

∫ X ϕ d Γ ( u , v ) = 1 2 [ ℰ ( ϕ u , v ) + ℰ ( ϕ v , u ) − ℰ ( ϕ , u v ) ] , ϕ ∈ ℱ ∩ C c ( X ) .

With respect to ℰ , the following intrinsic metric d ℰ on X is defined as follows:

d ℰ ( x , y ) = sup { u ( x ) − u ( y ) : u ∈ ℱ ∩ C 0 ( X ) and d Γ ( u , u ) ≤ d μ } ,

where the condition d Γ ( u , u ) ≤ d μ means that Γ ( u , u ) is absolutely continuous with respect to μ whose Radon-Nikodym derivative is bounded by 1.

Definition 1.1

A strongly local regular Dirichlet space is called strictly local if d ℰ is a metric on X and the topology induced by d ℰ coincides with the topology on X .

In the rest of this article, we always assume that the Dirichlet space ( X , μ , ℰ , ℱ ) is strictly local, so ℰ is strongly local and regular and d ℰ is a metric on X that induces the topology on X . The metric space X is complete. If Γ ( f , f ) is absolutely continuous with respect to μ , as is the case for locally Lipschitz functions, then ∣ ∇ f ∣ is the square root of its Radon-Nikodym derivative, so Γ ( f , f ) = ∣ ∇ f ∣ 2 d μ .

It should be pointed out that, in the previous references (cf. [8,9,25,28,33]), the metric measure space X satisfies the weak ( 1 , 1 ) -Poincaré inequality: there are constants C > 0 and λ ≥ 1 such that whenever B = B ( x , r ) = { y ∈ X : d ℰ ( x , y ) < r } is a ball in X ,

(3) 1 μ ( B ) ∫ B ∣ u − u B ∣ d μ ≤ C rad ( B ) μ ( λ B ) ∫ λ B ∣ ∇ u ∣ d μ ,

where rad ( B ) = r denotes the radius of the ball B and λ B = B ( x , λ r ) . In the aforementioned articles, the relative isoperimetric inequality deduced from equation (3) plays an important role. In this article, we assume that ( X , μ , ℰ , ℱ ) supports the weak ( 1 , 2 ) -Poincaré inequality, i.e.,

(4) 1 μ ( B ) ∫ B ∣ u − u B ∣ d μ ≤ C rad ( B ) 1 μ ( λ B ) ∫ λ B ∣ ∇ u ∣ 2 d μ 1 ∕ 2 .

From Hölder’s inequality, it is obvious that (4) is weaker than (3). With the help of properties of the perimeter and the heat semigroup on the Dirichlet space, we overcome the difficulties without the relative isoperimetric inequality, see the proofs of Theorems 3.1 and 4.4 for the details.

Moreover, we further assume that the Dirichlet space ( X , μ , ℰ , ℱ ) satisfies the following assumptions.

Assumption I: the doubling condition. We assume that μ is volume doubling throughout this article and its definition is given as follows.

Definition 1.2

We say that the metric measure space ( X , μ , ℰ , ℱ ) satisfies the volume doubling property if there exists a constant C > 0 such that for every x ∈ X and r > 0 ,

μ ( B ( x , 2 r ) ) ≤ C μ ( B ( x , r ) ) .

The above doubling property implies that there exist Q > 0 and C Q > 0 depending only on C in Definition 1.2 such that for every x ∈ X and all 0 < r < R ,

(5) μ ( B ( x , R ) ) ≤ C Q ( R ∕ r ) Q μ ( B ( x , r ) ) .

Note that ( X , μ , ℰ , ℱ ) satisfies the doubling property if and only if it satisfies equation (5) for some Q > 0 . Moreover, if equation (5) holds true for Q , then it holds true for each Q ˜ > Q . Hence, we shall assume without loss of generality that Q ≥ 2 .

Assumption II: the Poincaré inequality. The Dirichlet space ( X , μ , ℰ , ℱ ) also needs to support the following weak ( 1 , 2 ) -Poincaré inequality, i.e.,

Definition 1.3

Let 1 ≤ p < ∞ . We say that ( X , μ , ℰ , ℱ ) supports a weak ( 1 , p ) -Poincaré inequality if there are constants C > 0 and λ ≥ 1 such that whenever B is a ball in X (with respect to the metric d ℰ ) and u ∈ ℱ , we have

1 μ ( B ) ∫ B ∣ u − u B ∣ d μ ≤ C rad ( B ) 1 μ ( λ B ) ∫ λ B ∣ ∇ u ∣ p d μ 1 ∕ p .

Remark 1.4

Clearly, the Hölder inequality implies that the requirement that ( X , μ , ℰ , ℱ ) supports a weak ( 1 , 1 ) -Poincaré inequality is a significantly stronger requirement than supporting a weak ( 1 , 2 ) -Poincaré inequality. It should be noted that we need to assume that ( X , μ , ℰ , ℱ ) supports the weak ( 1 , 2 ) -Poincaré inequality in Sections 2.2, 3, and 4.

It follows from the study by Alonso-Ruiz et al. [2] that there are some examples of strictly local Dirichlet spaces ( X , μ , ℰ , ℱ ) that satisfy the volume doubling property and support the weak ( 1 , 2 ) -Poincaré inequality including:

Complete Riemannian manifolds with nonnegative Ricci curvature or, more generally, the R C D ( 0 , ∞ ) spaces in the sense of Ambrosio et al. (cf. [10]).
Carnot groups and other complete sub-Riemannian manifolds satisfying a generalized curvature dimension inequality (cf. [14,17]).
Doubling metric measure spaces that support a weak ( 1 , 2 ) -Poincaré inequality with respect to the upper gradient structure of Heinonen and Koskela (cf. [26,30,31]).
Metric graphs with bounded geometry (cf. [23]).

Assumption III: the curvature condition. In order to obtain our capacitary inequalities, we will need an additional requirement called the weak Bakry-Émery curvature condition.

Definition 1.5

We say that the Dirichlet metric space ( X , μ , ℰ , ℱ ) satisfies a weak Bakry-Émery curvature condition if, whenever u ∈ ℱ ∩ L ∞ ( X ) and t > 0 ,

(6) ‖ ∣ ∇ P t u ∣ ‖ L ∞ ( X ) 2 ≤ C t ‖ u ‖ L ∞ ( X ) 2 ,

where { P t } t ∈ [ 0 , ∞ ) is the self-adjoint semigroup of contractions on L 2 ( X , μ ) associated with the Dirichlet space ( X , μ , ℰ , ℱ ) .

The semigroup { P t } t ∈ [ 0 , ∞ ) is also referred to as the heat semigroup on ( X , μ , ℰ , ℱ ) , and it admits a heat kernel function p t ( x , y ) on [ 0 , ∞ ) × X × X for which there are positive constants c 1 , c 2 , and C such that whenever t > 0 and x , y ∈ X ,

(7) 1 C e − c 1 d ℰ ( x , y ) 2 ∕ t μ ( B ( x , t ) ) ≤ p t ( x , y ) ≤ C e − c 2 d ℰ ( x , y ) 2 ∕ t μ ( B ( x , t ) )

(cf. [41,42] or [2]). Moreover, following from [2], we know that the weak Bakry-Émery curvature condition is satisfied in the following examples:

Complete Riemannian manifolds with nonnegative Ricci curvature and, more generally, the R C D ( 0 , + ∞ ) spaces (cf. [27]).
Carnot groups (cf. [15]).
Complete sub-Riemannian manifolds with generalized nonnegative Ricci curvature (cf. [14,17]).
On noncompact metric graphs with finite number of edges, the weak Bakry-Émery curvature condition has been proved to hold for t ∈ ( 0 , 1 ] (cf. [16, Theorem 5.4]). If the graph is moreover compact, the weak Bakry-Émery estimate holds for every t > 0 (cf. [16, Theorem 5.4]).

The structure of this article is organized as follows: in Section 2, we recall the definition of the BV function and give the De Giorgi characterization of total variation under the quasi-Bakry-Émery curvature condition, and we also investigate the BV capacity and some related results on the Dirichlet space ( X , μ , ℰ , ℱ ) ; Section 3 gives connections between the Hausdorff measure and the Hausdorff capacity on the Dirichlet space ( X , μ , ℰ , ℱ ) ; in Section 4.1, we split the Sobolev inequality and isoperimetric inequality when we assume that the weak Bakry-Émery estimate (6) is satisfied; Section 4.2 gives the proof of Maz’ya-Sobolev inequality on the Dirichlet space ( X , μ , ℰ , ℱ ) .

Throughout this article, we will use C to denote the positive constants, which are independent of main parameters and may be different at each occurrence.

2 BV spaces and BV capacities on the Dirichlet space

2.1 BV spaces on the Dirichlet space

We will recall some results for BV spaces on the Dirichlet space established in [2]. For BV spaces on the metric space, please refer to [7–9,11,25,28,33,35,39]. Denote by Lip loc ( U ) the space of all local Lipschitz functions on U ⊆ X .

Definition 2.1

We say that u ∈ L 1 ( X ) belongs to BV ( X ) if there is a sequence of functions u k ∈ L 1 ( X ) ⋂ Lip loc ( X ) such that u k → u in L 1 ( X ) and

‖ D u ‖ ( X ) = liminf k → ∞ ∫ X ∣ ∇ u k ∣ d μ < ∞ .

The BV norm is given as follows:

‖ u ‖ BV ( X ) ≔ ‖ u ‖ L 1 ( X ) + ‖ D u ‖ ( X ) .

To our knowledge, the space BV ( X ) is not necessarily a Banach space unless the additional conditions are added.

Definition 2.2

For u ∈ BV ( X ) and open sets U ⊂ X , we set

‖ D u ‖ ( U ) = inf liminf k → ∞ ∫ U ∣ ∇ u k ∣ d μ : u k ∈ Lip loc ( U ) , u k → u in L 1 ( U ) ,

where the infimum is taken over all sequences

{ u k } k ⊂ Lip loc ( U ) converging to u in L 1 ( U ) .

Remark 2.3

If u ∈ Lip loc ( U ) , then we take u k = u in Definition 2.2 to obtain

‖ D u ‖ ( U ) ≤ ∫ U ∣ ∇ u ∣ d μ .

Furthermore, following from [25, Theorem 2.7] or [39, Section 3], for any open set U ⊆ X , we can choose a sequence of functions u k ∈ L 1 ( U ) ⋂ Lip loc ( U ) such that u k → u in L 1 ( U ) and ∫ U ∣ ∇ u k ∣ d μ → ‖ D u ‖ ( U ) as k → ∞ .

Lemma 2.4

(Lemma 3.4 in [2]) Let U and V be two open subsets of X . If u ∈ BV ( X ) , then

‖ D u ‖ ( ∅ ) = 0 ,
‖ D u ‖ ( U ) ≤ ‖ D u ‖ ( V ) if U ⊂ V ,
‖ D u ‖ ( ⋃ i = 1 ∞ U i ) = ∑ i = 1 ∞ ‖ D u ‖ ( U i ) if { U i } i is a pairwise disjoint subfamily of open subsets of X .

We will see in Definition 2.5 that ‖ D u ‖ can be extended from the collection of open sets to the collection of all Borel sets as a Radon measure.

Definition 2.5

For A ⊂ X , we set

‖ D u ‖ * ( A ) ≔ inf { ‖ D u ‖ ( O ) : O is an open subset of X , A ⊂ O } .

By the second property listed in Lemma 2.4, we note that if A is an open subset of X , then ‖ D u ‖ * ( A ) = ‖ D u ‖ ( A ) . With this observation, we rename ‖ D u ‖ * ( A ) as ‖ D u ‖ ( A ) even when A is not open.

Theorem 2.6

(Theorem 3.7 in [2]) If f ∈ BV ( X ) , then ‖ D f ‖ is a Radon outer measure on X , and the restriction of ‖ D f ‖ to the Borel sigma algebra is a Radon measure, which is the weak limit of ‖ D u k ‖ for some sequence { u k } of locally Lipschitz functions in L 1 ( X ) such that u k → f in L 1 ( X ) .

Definition 2.7

A function u is said to be in B V loc ( X ) if for each bounded open set U ⊆ X , there is a compactly supported Lipschitz function η U on X such that η U = 1 on U and η U u ∈ B V ( X ) . We say that a measurable set E ⊆ X is of finite perimeter if 1 E ∈ B V loc ( X ) with ‖ D 1 E ‖ ( X ) < ∞ . For any Borel set A ⊆ X , we denote by P ( E , A ) ≔ ‖ D 1 E ‖ ( A ) the perimeter measure of E .

Proposition 2.8

(Theorem 3.9 in [2]) The co-area formula on X holds true, i.e., for any Borel set A ⊆ X and u ∈ L loc 1 ( X ) ,

(8) ‖ D u ‖ ( A ) = ∫ R P ( { x ∈ X : u ( x ) > s } , A ) d s .

Moreover, the following max-min property of the variation has been proved in the study by Hakkarainen and Shanmugalingam [25].

Proposition 2.9

(Theorem 2.8 in [25]) Let u , v ∈ L 1 ( X ) . Then

‖ D max { u , v } ‖ ( X ) + ‖ D min { u , v } ‖ ( X ) ≤ ‖ D u ‖ ( X ) + ‖ D v ‖ ( X ) .

For any compact subsets E and F in X , by choosing u = 1 E and u = 1 F , Corollary 2.10 can be immediately deduced from Proposition 2.9.

Corollary 2.10

For any subsets E and F in X , we have

P ( E ∩ F , X ) + P ( E ∪ F , X ) ≤ P ( E , X ) + P ( F , X ) .

2.2 Total variation: De Giorgi characterization under the quasi-Bakry-Émery curvature condition

In this subsection, we prove a Dirichlet space version of the De Giorgi characterization in [22] of the total variation of a BV function under the following quasi-Bakry-Émery curvature condition.

Definition 2.11

We say that the Dirichlet metric space ( X , μ , ℰ , ℱ ) satisfies a quasi-Bakry-Émery curvature condition if there exists a constant C > 0 such that for every u ∈ ℱ and t ≥ 0 , we have

(9) ∣ ∇ P t u ∣ ≤ C P t ∣ ∇ u ∣ , μ a.e.

It can be seen from the proof of [16, Theorem 3.3] that the quasi-Bakry-Émery curvature condition implies the weak Bakry-Émery curvature condition (6). We refer the reader to [13,16,20,21] for the examples where the quasi-Bakry-Émery estimate is satisfied.

Marola et al., in their study [38], obtained the metric space version of these characterizations under the Bakry-Émery condition, which is also denoted by the B E ( K , ∞ ) condition. Please refer to Definition 5.1 for the B E ( K , ∞ ) condition in their article. It follows from [38, Proposition 6.2] that the weak ( 1 , 1 ) -Poincaré inequality can be seen as an equivalent formulation for the B E ( K , ∞ ) condition. By the previous argument, we cannot obtain the direct connection between the quasi-Bakry-Émery curvature condition and the B E ( K , ∞ ) condition.

The main result of this subsection is the following theorem.

Theorem 2.12

Assume that X supports a weak ( 1 , 2 ) -Poincaré inequality. Let u ∈ L 1 ( X ) . If

(10) limsup t → 0 + ∫ X ∣ ∇ P t u ∣ d μ < ∞ ,

then u ∈ BV ( X ) . Conversely, assume that the weak ( 1 , 2 ) -Poincaré inequality and the quasi Bakry-Émery estimate (9) are satisfied. If u ∈ BV ( X ) , then equation (10) holds true.

Proof

Since u ∈ L 1 ( X ) , by the strong continuity of { P t } t ≥ 0 on L 1 ( X ) (cf. [1]), we have

‖ P t u − u ‖ L 1 → 0 as t → 0 .

Using [40, Theorem 5.4.12] and combining the doubling property with the weak ( 1 , 2 ) -Poincaré inequality, we can obtain the Hölder regularity of the heat kernel p t ( ⋅ , ⋅ ) . Then, P t u is local Lipschitz continuous. The definition of the BV function implies that

‖ D u ‖ ( X ) ≤ limsup t → 0 + ∫ X ∣ ∇ P t u ∣ d μ .

Then, we conclude that u ∈ BV ( X ) .

Conversely, let u ∈ BV ( X ) . We can choose a sequence of functions u k ∈ L 1 ( X ) ⋂ Lip loc ( X ) such that u k → u in L 1 ( X ) and ∫ X ∣ ∇ u k ∣ d μ → ‖ D u ‖ ( X ) as k → ∞ . Since { P t } t ≥ 0 is a contraction semigroup on L 1 ( X ) , combining the lower semicontinuity of the total variation with the quasi-Bakry-Émery curvature condition deduces that

‖ D u ‖ ( X ) ≤ limsup t → 0 + ∫ X ∣ ∇ P t u ∣ d μ ≤ limsup t → 0 + liminf k → ∞ ∫ X ∣ ∇ P t u k ∣ d μ ≤ C limsup t → 0 + liminf k → ∞ ∫ X P t ∣ ∇ u k ∣ d μ ≤ C liminf k → ∞ ∫ X ∣ ∇ u k ∣ d μ = C ‖ D u ‖ ( X ) ,

which completes the proof of this theorem.□

2.3 BV capacities on the Dirichlet space

In this subsection, we investigate the BV capacity on the Dirichlet space ( X , μ , ℰ , ℱ ) . The BV capacity on the metric space equipped with a doubling measure has been obtained in the study by Hakkarainen and Kinnunen [24].

Definition 2.13

For a set E ⊆ X , let A ( E , BV ( X ) ) be the class of admissible functions on X , i.e., function f ∈ BV ( X ) satisfying 0 ≤ f ≤ 1 and f = 1 in a neighborhood of E (an open set containing E ). The BV capacity of E is defined as follows:

cap ( E , BV ( X ) ) ≔ inf { ‖ D f ‖ ( X ) : f ∈ A ( E , BV ( X ) ) } .

We next give two characterizations for the BV capacity on the Dirichlet space. Similar results were previously given in the study by Hakkarainen and Shanmugalingam [25], but we include the proof here as well for the sake of completeness.

Theorem 2.14

A geometric description of the BV capacity of a set in the Dirichlet space: for any set K ⊆ X ,

(11) cap ( K , BV ( X ) ) = inf A P ( A , X ) ,

where the infimum is taken over all sets A ⊆ X such that K ⊆ int ( A ) .

Proof

For any set A ⊆ X with K ⊆ int ( A ) and P ( A , X ) < ∞ , we have 1 A ∈ A ( K , BV ( X ) ) , and hence

cap ( K , BV ( X ) ) ≤ P ( A , X ) .

Taking the infimum over all such sets A implies that

cap ( K , BV ( X ) ) ≤ inf A P ( A , X ) .

To establish the opposite inequality, we may assume that cap ( K , BV ( X ) ) < ∞ . Let ε > 0 and f ∈ A ( K , BV ( X ) ) such that

‖ D f ‖ ( X ) < cap ( K , BV ( X ) ) + ε .

By the co-area formula (8), we have

‖ D f ‖ ( X ) = ∫ 0 1 P ( { x ∈ X : f ( x ) > t } , X ) d t .

Then, there exists a t 0 ∈ ( 0 , 1 ) such that

P ( { x ∈ X : f ( x ) > t 0 } , X ) ≤ ‖ D f ‖ ( X ) < cap ( K , BV ( X ) ) + ε .

Noting that

K ⊆ int { x ∈ X : f ( x ) > t 0 } ,

then we have

inf { A , K ⊆ int ( A ) } P ( A , X ) < cap ( K , BV ( X ) ) + ε .

Consequently, equation (11) can be deduced by letting ε → 0 .□

Theorem 2.15

If K is a compact subset of X , then

cap ( K , BV ( X ) ) = inf ∫ X ∣ ∇ f ∣ d μ : f ∈ A ( K , BV ( X ) ) ∩ Lip c ( X ) .

Proof

Definition 2.13 implies that

cap ( K , BV ( X ) ) ≤ inf ∫ X ∣ ∇ f ∣ d μ : f ∈ A ( K , BV ( X ) ) ∩ Lip c ( X ) .

Conversely, given any ε ∈ ( 0 , 1 ) , there exists a function f ∈ A ( K , BV ( X ) ) such that

‖ D f ‖ ( X ) < cap ( K , BV ( X ) ) + ε ,

where f = 1 on the open set U ⊇ K with μ ( U ) < ∞ . Since K is compact and X \ U is a closed set, we can find an open set U ′ ⊆ U such that K ⊆ U ′ ⊆ U and dist X ( U ′ , X \ U ) > 0 , where

dist X ( U ′ , X \ U ) ≔ inf { d ℰ ( x , y ) : x ∈ U ′ , y ∈ X \ U } .

Remark 2.3 tells us that there exists a sequence of functions f i ∈ Lip c ( X ) ∩ BV ( X ) , i = 1 , 2 , … , such that f i → f in L 1 ( X ) and

∫ X ∣ ∇ f i ∣ d μ → ‖ D f ‖ ( X )

as i → ∞ . Let η ∈ Lip c ( X ) be a cutoff function such that 0 ≤ η ≤ 1 , η = 1 in U ′ , and η = 0 in X \ U . For every i = 1 , 2 , … , we define functions

v i = ( 1 − η ) f i + η ,

and note that v i ∈ Lip c ( X ) . In addition, v i = 1 in U ′ for every index i , and applying the Leibniz rule implies that

∇ v i = ( 1 − η ) ∇ f i + ( 1 − f i ) ∇ η .

Since f = ( 1 − η ) f + η , we obtain

lim sup i → ∞ ∫ X ∣ ∇ v i ∣ d μ ≤ lim sup i → ∞ ∫ X ∣ ( 1 − η ) ∇ f i ∣ d μ + lim sup i → ∞ ∫ X ∣ ( 1 − f i ) ∇ η ∣ d μ ≤ ‖ D f ‖ ( X ) + ‖ ∇ η ‖ ∞ lim sup i → ∞ ∫ U ∣ f − f i ∣ d μ < cap ( K , BV ( X ) ) + ε .

Therefore, for every ε > 0 , we can find a function v ∈ A ( K , BV ( X ) ) ∩ Lip c ( X ) such that

∫ X ∣ ∇ v ∣ d μ < cap ( K , BV ( X ) ) + ε .

By taking the infimum over such functions, we can obtain the reverse inequality.□

Finally, the following theorem gives the metric-theoretic properties of the BV capacity (cf. [24, Section 3]).

Proposition 2.16

Assume A and B are subsets of X .

cap ( ∅ , BV ( X ) ) = 0 .
If A ⊆ B , then
cap ( A , BV ( X ) ) ≤ cap ( B , BV ( X ) ) .
(12) cap ( A ∪ B , BV ( X ) ) + cap ( A ∩ B , BV ( X ) ) ≤ cap ( A , BV ( X ) ) + cap ( B , BV ( X ) ) ,
whose equality can be assured by the subadditivity above and the constraint below
(13) cap ( A \ ( A ∩ B ) , BV ( X ) ) ⋅ cap ( B \ ( B ∩ A ) , BV ( X ) ) = 0 .
If A k , k = 1 , 2 , … , are compact sets in X satisfying A 1 ⊇ A 2 ⊇ ⋯ ⊇ A k ⊇ ⋯ , then
lim k → ∞ cap ( A k , BV ( X ) ) = cap ⋂ k = 1 ∞ A k , BV ( X ) .
If A k , k = 1 , 2 , … , are sets in X , then
cap ⋃ k = 1 ∞ A k , BV ( X ) ≤ ∑ k = 1 ∞ cap ( A k , BV ( X ) ) .
For any sequence { A k } k = 1 ∞ of subsets of X with A 1 ⊆ A 2 ⊆ ⋯ ,
lim k → ∞ cap ( A k , BV ( X ) ) = cap ⋃ k = 1 ∞ A k , BV ( X ) .

Proof

Similarly to Xiao and Zhang’s results in [45], we only need to show that the equality of equation (12) is valid. Since equation (12) is valid, we only need to prove that its converse inequality holds true under the above condition (equation (13)). Obviously, the condition cap ( A \ ( A ∩ B ) , BV ( X ) ) ⋅ cap ( B \ ( B ∩ A ) , BV ( X ) ) = 0 implies that cap ( A \ ( A ∩ B ) , BV ( X ) ) = 0 or cap ( B \ ( B ∩ A ) , BV ( X ) ) = 0 . Suppose

cap ( A \ ( A ∩ B ) , BV ( X ) ) = 0 .

By equation (12), we have

(14) cap ( A , BV ( X ) ) = cap ( ( A ⧹ ( A ∩ B ) ) ∪ ( A ∩ B ) , BV ( X ) ) ≤ cap ( A ⧹ ( A ∩ B ) , BV ( X ) ) + cap ( A ∩ B , BV ( X ) ) = cap ( A ∩ B , BV ( X ) ) .

Using Definition 2.13 and B ⊆ A ∪ B , we obtain

(15) cap ( B , BV ( X ) ) = inf f ∈ A ( B , BV ( X ) ) { ‖ D f ‖ ( X ) } ≤ inf f ∈ A ( A ∪ B , BV ( X ) ) { ‖ D f ‖ ( X ) } = cap ( A ∪ B , BV ( X ) ) .

Combining equation (14) with equation (15), we obtain

cap ( A , BV ( X ) ) + cap ( B , BV ( X ) ) ≤ cap ( A ∩ B , BV ( X ) ) + cap ( A ∪ B , BV ( X ) ) ,

which derives the desired result. Another case can be similarly proved; we omit the details.□

Corollary 2.17

Assume E ⊆ X .

cap ( E , BV ( X ) ) = inf { cap ( O , BV ( X ) ) : open O ⊇ E } .
If E is a Borel set, then
cap ( E , BV ( X ) ) = sup { cap ( K , BV ( X ) ) : compact K ⊆ E } .

3 Connection between Hausdorff measure and Hausdorff capacity

In this section, let ( X , μ , ℰ , ℱ ) be a Dirichlet space supporting the weak ( 1 , 2 ) -Poincaré inequality. We stress that the weak ( 1 , 1 ) -Poincaré inequality is not assumed.

We first define the Hausdorff capacity of E ⊆ X on the metric space X as follows:

ℋ ∞ ( E ) ≔ inf ∑ i = 1 ∞ μ ( B i ) rad ( B i ) : E ⊆ ∪ i = 1 ∞ B i ,

where rad ( B i ) denotes the radius of the balls B i , i = 1 , 2 , … .

Moreover, the Hausdorff measure of E ⊆ X is defined as follows:

ℋ ( E ) ≔ sup δ > 0 ℋ δ ( E ) ,

where

ℋ δ ( E ) ≔ inf ∑ i = 1 ∞ μ ( B i ) rad ( B i ) : E ⊆ ∪ i = 1 ∞ B i , rad ( B i ) ≤ δ

with 0 < δ < ∞ . It is obvious that ℋ ∞ ( E ) ≤ ℋ ( E ) for any set E ⊆ X .

The following theorem reveals the equivalence between the BV capacity and the Hausdorff capacity on the Dirichlet space ( X , μ , ℰ , ℱ ) . The same equivalence has been obtained in [28] and [37] if the metric space X satisfies the stronger condition, i.e., X supports the weak ( 1 , 1 ) -Poincaré inequality.

Theorem 3.1

There exist two positive constants C 1 and C 2 such that

C 2 ℋ ∞ ( E ) ≤ cap ( E , BV ( X ) ) ≤ C 1 ℋ ∞ ( E )

for any compact set E ⊆ X .

Proof

First, we consider

cap ( E , BV ( X ) ) ≤ C 1 ℋ ∞ ( E ) .

Take any covering balls { B ( x i , r i ) } such that E ⊆ ⋃ i = 1 ∞ B ( x i , r i ) . By conditions (ii) and (v) of Proposition 2.16, we have

cap ( E , BV ( X ) ) ≤ cap ⋃ i = 1 ∞ B ( x i , r i ) , BV ( X ) ≤ ∑ i = 1 ∞ cap ( B ( x i , r i ) , BV ( X ) ) .

Applying the admissible function

ϕ i ( x ) = max 0 , 1 − dist X ( x , B ( x i , r i ) ) rad ( B ( x i , r i ) )

to Theorem 2.15, we have

cap ( B ( x i , r i ) , BV ( X ) ) ≤ C μ ( B ( x i , r i ) ) rad ( B ( x i , r i ) ) .

Then,

cap ( E , BV ( X ) ) ≤ C ∑ i = 1 ∞ μ ( B ( x i , r i ) ) rad ( B ( x i , r i ) ) .

Hence, the definition of ℋ ∞ ( E ) implies that

cap ( E , BV ( X ) ) ≤ C 1 ℋ ∞ ( E ) .

Second, we prove that C 2 ℋ ∞ ( E ) ≤ cap ( E , BV ( X ) ) . Take a bounded open set U in X containing E with μ ( U ) < ∞ . Let x ′ ∈ U . Since U is open, then there exists r x ′ > 0 such that

μ ( U ⋂ B ( x ′ , r x ′ ) ) μ ( B ( x ′ , r x ′ ) ) = 1 .

Since μ ( U ) < ∞ , we have

lim r → ∞ μ ( U ⋂ B ( x ′ , r ) ) μ ( B ( x ′ , r ) ) = 0 .

Hence, there exists a positive integer k such that

μ ( U ⋂ B ( x ′ , 2 m r x ′ ) ) μ ( B ( x ′ , 2 m r x ′ ) ) > 1 ∕ 2

for 0 ≤ m < k and

μ ( U ⋂ B ( x ′ , 2 k r x ′ ) ) μ ( B ( x ′ , 2 k r x ′ ) ) ≤ 1 ∕ 2 .

Therefore,

μ ( B ( x ′ , 2 k r x ′ ) \ U ) = μ ( B ( x ′ , 2 k r x ′ ) ) − μ ( B ( x ′ , 2 k r x ′ ) ⋂ U ) ≥ μ ( B ( x ′ , 2 k r x ′ ) ⋂ U ) .

The doubling condition of the measure μ implies that

μ ( U ⋂ B ( x ′ , 2 k r x ′ ) ) μ ( B ( x ′ , 2 k r x ′ ) ) ≥ μ ( U ⋂ B ( x ′ , 2 k − 1 r x ′ ) ) C μ ( B ( x ′ , 2 k − 1 r x ′ ) ) ≥ 1 2 C .

Then,

μ ( B ( x ′ , 2 k r x ′ i ) ⋂ U ) μ ( B ( x ′ , 2 k r x ′ i ) \ U ) ( μ ( B ( x ′ , 2 k r x ′ i ) ) ) 2 ≥ 1 4 C 2 > 0 .

Denote by 2 k r x ′ = R x ′ with k being defined as above. By [2, Theorem 4.4], we have

P ( U , B ( x ′ , R x ′ ) ) ≥ limsup t → 0 1 t ∫ B ( x ′ , R x ′ ) ∫ B ( x ′ , R x ′ ) p t ( x , y ) ∣ 1 E ( x ) − 1 E ( y ) ∣ d μ ( x ) d μ ( y ) ≥ limsup t → 0 1 t ∫ B ( x ′ , R x ′ ) ⋂ U ∫ B ( x ′ , R x ′ ) \ U p t ( x , y ) ∣ 1 E ( x ) − 1 E ( y ) ∣ d μ ( x ) d μ ( y ) ≥ e − C R x ′ μ ( B ( x ′ , R x ′ ) ⋂ U ) μ ( B ( x ′ , R x ′ ) \ U ) μ ( B ( x ′ , 2 R x ′ ) ) ≥ C μ ( B ( x ′ , R x ′ ) ) R x ′ ≥ C μ ( B ( x ′ , R x ′ ) ) rad ( B ( x ′ , R x ′ ) ) ,

where we have used equation (7), which is the lower bound of heat kernel p t ( x , y ) , in the third step.

Applying a general 5-covering lemma (cf. [28, Theorem 3.1]) to the family of balls { B ( x ′ , R x ′ ) } for x ′ ∈ U , we know that there exists a disjoint collection of balls { B ( x i , c R i ) } with x i ∈ U for c ≥ 1 and

U ⊆ ⋃ x ′ ∈ U ∞ B ( x ′ , c R x ′ ) ⊆ ⋃ i = 1 ∞ B ( x i , 5 c R i ) .

Therefore, by Lemma 2.4, we have

∑ i = 1 ∞ μ ( B ( x i , 5 c R i ) ) rad ( B ( x i , 5 c R i ) ) ≤ C ∑ i = 1 ∞ μ ( B ( x i , c R i ) ) rad ( B ( x i , c R i ) ) ≤ C ∑ i = 1 ∞ P ( U , B ( x i , c R i ) ) = C P U , ⋃ i = 1 ∞ B ( x i , c R i ) ≤ C P ( U , X ) .

Let ε > 0 . It follows from Theorem 2.15 that there exists a function f ∈ A ( E , BV ( X ) ) ∩ Lip c ( X ) such that

∫ X ∣ ∇ f ∣ d μ < cap ( E , BV ( X ) ) + ε .

By contradiction, the co-area formula in Proposition 2.8, and Remark 2.3, we conclude that there exists a s 0 ∈ ( 0 , 1 ) such that

P ( { x ∈ X : f ( x ) > s 0 } , X ) ≤ ∫ R P ( { x ∈ X : f ( x ) > s } , X ) d s = ‖ D f ‖ ( X ) ≤ ∫ X ∣ ∇ f ∣ d μ < cap ( E , BV ( X ) ) + ε .

It is easy to see that the set { x ∈ X : f ( x ) > s 0 } is an open neighborhood of E . By the monotonicity of Hausdorff capacity, we have

ℋ ∞ ( E ) ≤ ℋ ∞ ( U ) ≤ C P ( U ) ≤ C ( cap ( E , BV ( X ) ) + ε ) .

Letting ε → 0 implies that C 2 ℋ ∞ ( E ) ≤ cap ( E , BV ( X ) ) holds true.□

Lemma 3.2

Assume that ( X , μ , ℰ , ℱ ) is a Dirichlet space supporting the weak ( 1 , 2 ) -Poincaré inequality, and for any ball B ( x , r ) in X , the volume growth condition μ ( B ( x , r ) ) ≥ C 3 r Q is satisfied for some Q ≥ 1 . For any set E ⊂ X , if ℋ ∞ ( E ) = 0 , then ℋ ( E ) = 0 .

Proof

Since ℋ ∞ ( E ) = 0 , and for any δ > 0 , we have ℋ ∞ ( E ) = min { ℋ δ ( E ) , S } , where

S = inf ∑ i = 1 ∞ μ ( B i ) rad ( B i ) : E ⊆ ⋃ i = 1 ∞ B i , sup rad ( B i ) > δ .

It is easy to see that S > C 3 δ Q − 1 > 0 , then we have ℋ δ ( E ) = 0 . Therefore, ℋ ( E ) = sup δ > 0 ℋ δ ( E ) = 0 .□

The following corollary reveals that the BV capacity and the Hausdorff measure have the same null sets. We easily prove it by Theorem 3.1, Lemma 3.2 and measure-theoretic properties of the BV capacity.

Corollary 3.3

Assume that ( X , μ , ℰ , ℱ ) is a Dirichlet space supporting the weak ( 1 , 2 ) -Poincaré inequality, and for any ball B ( x , r ) in X , the volume growth condition μ ( B ( x , r ) ) ≥ C 3 r Q is satisfied for some Q ≥ 1 . For any set E ⊆ X , then ℋ ( E ) = 0 if and only if cap ( E , BV ( X ) ) = 0 .

4 Several capacitary inequalities on the Dirichlet space

4.1 BV isocapacity inequalities on the Dirichlet space

Alonso-Ruiz et al., in their study [2] obtained the following Sobolev inequality and isoperimetric inequality in the metric measure Dirichlet setting under the further hypothesis that a weak Bakry-Émery estimate is satisfied. In this subsection, we generalize Xiao’s results in [43] to the Euclidean space, i.e., we split the Sobolev inequality and isoperimetric inequality on the Dirichlet space when we assume that X supports a weak ( 1 , 2 ) -Poincaré inequality and the weak Bakry-Émery estimate (6) is satisfied. At this time, [1, Proposition 4.14] and [2, Theorem 4.4] imply that BV ( X ) is a Banach space.

Moreover, we need to recall the Minkowski content on the metric space, which has a close relation with the perimeter (see [12] and the references therein). The upper Minkowski content is defined by

ℳ − ( M ) ≔ liminf r → 0 + μ ( M r ) − μ ( M ) r

for Borel sets M ⊆ X with finite μ -measure, where M r ≔ { x ∈ X : dist X ( x , M ) < r } . Ambrosio et al., in their study [12], also defined the relaxed Minkowski contents as follows:

ℳ − * ( M ) ≔ inf { liminf h → ∞ ℳ − ( M h ) : M h → M in μ -measure } ,

where { M h } h = 1 ∞ are Borel sets with finite μ -measure.

Lemma 4.1

(Proposition 4.9 in [2]) Let the Dirichlet space ( X , μ , ℰ , ℱ ) be a complete doubling metric measure space supporting the weak ( 1 , 2 ) -Poincaré inequality and the weak Bakry-Émery estimate (6). For any Borel set M ⊆ X with μ ( M ) < ∞ , one has

P ( M , X ) ≤ ℳ − ( M ) .

Under the regularity assumption

(16) ‖ D f ‖ ( X ) = ∫ X ∣ ∇ f ∣ d μ

for any function f ∈ L 1 ( X ) ⋂ Lip ( X ) and μ ( { x ∈ X : f ( x ) > 0 } r ) < ∞ , the co-area formula in equation (8) can be replaced by the another form, which is proved in [12, Proposition 4.3], i.e.,

(17) ‖ D f ‖ ( X ) = ∫ 0 ∞ ℳ − ( { x ∈ X : f ( x ) ≥ t } ) d t

for any nonnegative Lipschitz function f on X with μ ( { x ∈ X : f ( x ) > 0 } r ) < ∞ for some r > 0 .

Proposition 4.2

Let the Dirichlet space ( X , μ , ℰ , ℱ ) be a complete doubling metric measure space supporting the weak ( 1 , 2 ) -Poincaré inequality and the weak Bakry-Émery estimate (6). If the volume growth condition μ ( B ( x , r ) ) ≥ C 3 r Q , r > 0 , is satisfied for some Q > 0 , then there exists a positive constant C 4 such that for every f ∈ B V ( X ) ,

(18) ‖ f ‖ L q ( X ) ≤ C 4 ‖ D f ‖ ( X ) ,

where q = Q ∕ ( Q − 1 ) . In particular, if E is a Borel set with finite perimeter in X , then

[ μ ( E ) ] ( Q − 1 ) ∕ Q ≤ C 4 ℳ − ( E ) .

Proof

[2, Theorem 5.3] and Lemma 4.1 deduce the proposition.□

Theorem 4.3

Let the Dirichlet space ( X , μ , ℰ , ℱ ) be a complete doubling metric measure space. Then, the following two assertions are equivalent:

There exists a positive constant C 4 such that for any compact set M in X ,
(19) [ μ ( M ) ] ( Q − 1 ) ∕ Q ≤ C 4 cap ( M , BV ( X ) ) .
There exists a positive constant C 4 such that for any f ∈ L c Q ∕ ( Q − 1 ) ( X ) ,
(20) ‖ f ‖ Q ∕ ( Q − 1 ) ≤ C 4 ∫ 0 ∞ ( cap ( { x ∈ X : ∣ f ( x ) ∣ ≥ t } , BV ( X ) ) ) Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) ( Q − 1 ) ∕ Q .

Moreover, assume that the Dirichlet space ( X , μ , ℰ , ℱ ) supports the weak ( 1 , 2 ) -Poincaré inequality and the weak Bakry-Émery estimate (6), and the volume growth condition as in Proposition 4.2 is valid. Then, the inequalities (19) and (20) are true.

Proof

Part 1: We prove that assertions (i) and (ii) are equivalent.

(ii) ⇒ (i) First, we always adopt the short notation:

Ω t ( f ) = { x ∈ X : ∣ f ( x ) ∣ ≥ t }

for a function f defined on X and a number t > 0 . Given a compact set M ⊆ X , let f = 1 M . Then,

‖ f ‖ Q ∕ ( Q − 1 ) = [ μ ( M ) ] ( Q − 1 ) ∕ Q

and

Ω t ( f ) = M , if t ∈ ( 0 , 1 ] ; ∅ , if t ∈ ( 1 , ∞ ) .

Hence,

∫ 0 ∞ ( cap ( Ω t ( f ) , BV ( X ) ) ) Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) = ∫ 0 1 ( cap ( Ω t ( f ) , BV ( X ) ) ) Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) + ∫ 1 ∞ ( cap ( Ω t ( f ) ) , BV ( X ) ) Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) = ( cap ( M , BV ( X ) ) ) Q ∕ ( Q − 1 ) .

Therefore, equation (20) implies equation (19), i.e.,

(ii) ⇒ (i) .

(i) ⇒ (ii) We show that equation (19) implies equation (20). Suppose equation (19) holds for any compact set in X . For t > 0 and f ∈ L c Q ∕ ( Q − 1 ) ( X ) , using the inequality (19), we have

‖ f ‖ Q ∕ ( Q − 1 ) Q ∕ ( Q − 1 ) = ∫ 0 ∞ μ ( Ω t ( f ) ) d t Q ∕ ( Q − 1 ) ≤ C 4 ∫ 0 ∞ [ cap ( Ω t ( f ) , BV ( X ) ) ] Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) ,

which deduces that (i) ⇒ (ii) .

Part 2: Since we assume that the weak Bakry-Émery estimate (6) and the volume growth condition as in Proposition 4.2 are valid, then by the definition of cap ( ⋅ , BV ( X ) ) to equation (18) and Theorem 2.15, we have

[ μ ( M ) ] ( Q − 1 ) ∕ Q ≤ C 4 inf { ‖ D f ‖ ( X ) : f ∈ A ( M , BV ( X ) ) ∩ Lip c ( X ) } ,

namely, equation (19) holds true. Since equation (20) is equivalent to equation (19), so equation (20) is also true.□

In order to give the following theorem, we need to consider a class of smooth domain in the setting of metric space, which is called the uniform domain, i.e., a domain Ω ⊆ X is said to be A -uniform, with constant A ≥ 1 , if for every x , y ∈ Ω , there exists a curve γ in Ω connecting x and y such that l γ ≤ A d ℰ ( x , y ) , and for all t ∈ [ 0 , l γ ] , we have

dist X ( γ ( t ) , X \ Ω ) ≥ A − 1 min { t , l γ − t } ,

where l γ is the length of a curve γ .

The uniform domain is a natural generalization of Euclidean domains with Lipschitz boundary and has been investigated in [19,32,33]. They play an important role in the research of extension properties of functions of bounded variation and Newton-Sobolev functions on metric spaces.

Theorem 4.4

For all uniform compact domains M in X ,
(21) cap ( M , BV ( X ) ) ≤ ℳ − ( M ) .
For all functions f ∈ L i p c ( X ) ,
(22) ∫ 0 ∞ [ cap ( { x ∈ X : ∣ f ( x ) ∣ ≥ t } , BV ( X ) ) ] Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) ( Q − 1 ) ∕ Q ≤ ‖ D f ‖ ( X ) .

Proof

(ii) ⇒ (i) We only need to verify the equivalence between equations (22) and (21). For δ > 0 and a uniform compact domain M ⊆ X , define the Lipschitz function

f δ ( x ) ≔ 1 − dist X ( x , M ) δ , dist X ( x , M ) < δ ; 0 , dist X ( x , M ) ≥ δ .

If equation (22) holds, since M ⊆ Ω t ( f δ ) for t ∈ [ 0 , 1 ] , an application of Theorem 2.15 deduces that

cap ( M , BV ( X ) ) ≤ ∫ 0 1 ( cap ( Ω t ( f δ ) , BV ( X ) ) ) Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) ( Q − 1 ) ∕ Q ≤ ‖ D f δ ‖ ( X ) .

Using Remark 2.3, we have

‖ D f δ ‖ ( X ) ≤ ∫ X ∣ ∇ f δ ∣ d μ ≤ 1 δ ∫ { x ∈ X : 0 < dist X ( x , M ) < δ } d μ ,

which implies lim inf δ → 0 + ‖ D f δ ‖ ( X ) ≤ ℳ − ( M ) . Thus, equation (21) is obtained. Then, we conclude that (ii) ⇒ (i) .

(i) ⇒ (ii) Assume that equation (21) holds true for all uniform compact domains M with finite perimeter in X . The monotonicity of cap ( ⋅ , BV ( X ) ) implies that t → cap ( Ω t ( f ) , BV ( X ) ) is a decreasing function on [ 0 , ∞ ) . Then,

t 1 ∕ ( Q − 1 ) cap ( Ω t ( f ) , BV ( X ) ) Q ∕ ( Q − 1 ) = ( t cap ( Ω t ( f ) , BV ( X ) ) ) 1 ∕ ( Q − 1 ) cap ( Ω t ( f ) , BV ( X ) ) ≤ ∫ 0 t cap ( Ω r ( f ) , BV ( X ) ) d r 1 ∕ ( Q − 1 ) cap ( Ω t ( f ) , BV ( X ) ) = ( 1 − 1 ∕ Q ) d d t ∫ 0 t cap ( Ω r ( f ) , BV ( X ) ) d r Q ∕ ( Q − 1 ) .

Combining equation (21) with the above estimate deduces that

∫ 0 ∞ ( cap ( Ω t ( f ) , BV ( X ) ) ) Q ∕ ( Q − 1 ) d t Q ∕ ( Q − 1 ) = Q ∕ ( Q − 1 ) ∫ 0 ∞ ( cap ( Ω t ( f ) , BV ( X ) ) ) Q ∕ ( Q − 1 ) t 1 d − 1 d t ≤ ∫ 0 ∞ d d t ∫ 0 t cap ( Ω r ( f ) , BV ( X ) ) d r Q ∕ ( Q − 1 ) d t = ∫ 0 ∞ cap ( Ω t ( f ) , BV ( X ) ) d t Q ∕ ( Q − 1 ) ≤ ∫ 0 ∞ ℳ − ( Ω t ( f ) ) d t Q ∕ ( Q − 1 ) = ( ‖ D f ‖ ( X ) ) Q ∕ ( Q − 1 ) ,

where we have used the co-area formula (17) in the last step.□

Remark 4.5

In a recent article [18], under the weaker assumptions, Coulhon et al. obtained the isoperimetric inequality on the Dirichlet space, see [18, Theorems 6.3] for more details. It is natural to ask whether Theorems 4.3 and 4.4 are also valid under the conditions proposed in [18, Theorem 6.3]. We will consider these questions in a forthcoming article.

4.2 Maz’ya-Sobolev inequalities on the Dirichlet space

On the metric spaces equipped with a doubling measure supporting a weak ( 1 , 1 ) -Poincaré inequality, the Maz’yz-Sobolev inequalities had been proved in [33]. In this subsection, we obtain similar result on the Dirichlet space via replacing a weak ( 1 , 1 ) -Poincaré inequality by a weak ( 1 , 2 ) -Poincaré inequality. Moreover, the weak Bakry-Émery estimate (6) is also satisfied. Different from [33, Theorem 7.2], we obtain the Maz’ya-Sobolev inequalities related to the zero set of the BV function instead of its jump set.

We are now in a position to give the main result in this subsection.

Theorem 4.6

Let the Dirichlet space ( X , μ , ℰ , ℱ ) be a complete doubling metric measure space supporting the weak ( 1 , 2 ) -Poincaré inequality and the weak Bakry-Émery estimate (6). Assume the volume growth condition as in Proposition 4.2 is valid. For u ∈ BV ( X ) ⋂ Lip ( X ) , let S ≔ { x ∈ X : u ( x ) = 0 } . Then, for every B = B ( x , r ) ⊆ X , we have

1 μ ( 2 B ) ∫ 2 B ∣ u ∣ Q ∕ ( Q − 1 ) d μ ( Q − 1 ) ∕ Q ≤ C cap ( S ⋂ B , BV ( X ) ) ‖ D u ‖ ( 2 B ) ,

where Q is the index appearing in Proposition 4.2.

Proof

First, we assume that u > 0 . Let η : X → [ 0 , 1 ] be a ( μ ( 2 B ) ) − 1 ∕ Q -Lipschitz function with η = 1 in B and η = 0 in X \ 2 B . Denote by

a ≔ 1 μ ( 2 B ) ∫ 2 B u ( x ) Q ∕ ( Q − 1 ) d μ ( Q − 1 ) ∕ Q

and define ϕ ≔ η ( 1 − u ∕ a ) . It is easy to see that ϕ ∈ BV ( X ) such that ϕ = 0 in X \ 2 B and ϕ = 1 on S ⋂ B . Take an arbitrary compact set K ⊆ S ⋂ B such that ϕ = 1 on K . Employing Corollary 2.17 and the Leibniz rule yields

cap ( K , BV ( X ) ) ≤ C ∫ X ∣ ∇ ϕ ( x ) ∣ d μ ≤ C a ‖ D u ‖ ( 2 B ) + ∫ 2 B ∣ u − a ∣ ∣ ∇ η ∣ d μ = C a ‖ D u ‖ ( 2 B ) + C a ∫ 2 B ∣ u − a ∣ ∣ ∇ η ∣ d μ .

An application of Proposition 4.2 to function u , together with the Hölder inequality, tells us that

∫ 2 B ∣ u − a ∣ ∣ ∇ η ∣ d μ ≤ ∫ 2 B ∣ u ∣ ∣ ∇ η ∣ d μ + a μ ( 2 B ) ( μ ( 2 B ) ) − 1 ∕ Q ≤ ‖ D u ‖ ( 2 B ) .

Then, we have

cap ( K , BV ( X ) ) ≤ C a ‖ D u ‖ ( 2 B ) .

Therefore, (ii) of Corollary 2.17 implies that

1 μ ( 2 B ) ∫ 2 B ∣ u ∣ Q ∕ ( Q − 1 ) d μ ( Q − 1 ) ∕ Q ≤ C cap ( S ⋂ B , BV ( X ) ) ‖ D u ‖ ( 2 B ) .

For any set A ⊆ X , the definition of ‖ D u ‖ ( ⋅ ) implies

‖ D ∣ u ∣ ‖ ( A ) ≤ ‖ D u ‖ ( A ) .

So we can allow u to take negative value in the previous argument. Therefore, this completes the proof of this theorem.□

Finally, we use Theorem 4.6 to prove the following corollary, which is the result of [29, Lemma 2.2] for the case of Lipschitz functions, where the authors prove their result using the Sobolev-Poincaré inequality (cf. [29, Section 2]).

Corollary 4.7

Let the Dirichlet space ( X , μ , ℰ , ℱ ) be a complete doubling metric measure space supporting the weak ( 1 , 2 ) -Poincaré inequality and the weak Bakry-Émery estimate (6). Assume that the volume growth condition as in Proposition 4.2 is valid. For u ∈ BV ( X ) ⋂ Lip ( X ) , let S ≔ { x ∈ X : u ( x ) = 0 } . If μ ( B \ ( S ⋂ B ) ) ≤ γ μ ( B ) for some 0 < γ < 1 , then

1 μ ( B ) ∫ B ∣ u ∣ Q ∕ ( Q − 1 ) d μ ( Q − 1 ) ∕ Q ≤ C ( 1 − γ 1 ∕ Q ) − 1 μ ( B ) 1 ∕ Q ‖ D u ‖ ( 2 B ) μ ( 2 B ) .

Proof

Since

μ ( S ⋂ B ) = μ ( B ) − μ ( B \ ( S ⋂ B ) ) ≥ ( 1 − γ ) μ ( B ) ,

by Theorem 4.6 and equation (19), we have

1 μ ( B ) ∫ B ∣ u ∣ Q ∕ ( Q − 1 ) d μ ( Q − 1 ) ∕ Q ≤ C μ ( S ⋂ B ) cap ( S ⋂ B , BV ( X ) ) ‖ D u ‖ ( 2 B ) μ ( S ⋂ B ) ≤ C ( μ ( S ⋂ B ) ) 1 ∕ Q ‖ D u ‖ ( 2 B ) μ ( S ⋂ B ) ≤ C ( 1 − γ ) 1 ∕ Q − 1 μ ( B ) 1 ∕ Q ‖ D u ‖ ( 2 B ) μ ( 2 B ) ≤ C ( 1 − γ 1 ∕ Q ) − 1 μ ( B ) 1 ∕ Q ‖ D u ‖ ( 2 B ) μ ( 2 B ) ,

which completes the proof of this corollary.□

Acknowledgements

The authors are grateful to the anonymous reviewers for careful reading and valuable comments that helped to improve the article.

Funding information: Y. Liu was supported by the Beijing Natural Science Foundation of China (No. 1232023), the National Natural Science Foundation of China (No. 12271042), and the National Science and Technology Major Project of China (Nos J2019-I-0019-0018 and J2019-I-0001-0001). P.T. Li was supported by the National Natural Science Foundation of China (No. 11871293), the Shandong Natural Science Foundation of China (No. ZR2017JL008), and the University Science and Technology Projects of Shandong Province (No. J15LI15). J.Z. Huang was supported by the Fundamental Research Funds for the Central Universities (No. 500419772).
Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2023-04-09

Revised: 2023-09-06

Accepted: 2024-01-09

Published Online: 2024-02-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2023-0119

Keywords for this article

BV space; Dirichlet space; heat kernel; capacity; Sobolev inequality

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