Bourgain-Brezis-Mironescu approach in metric spaces with Euclidean tangents
aa r X i v : . [ m a t h . M G ] A p r BOURGAIN-BREZIS-MIRONESCU APPROACH IN METRIC SPACESWITH EUCLIDEAN TANGENTS
WOJCIECH G ´ORNY
Abstract.
In the setting of metric measure spaces satisfying the doubling conditionand the (1 , p )-Poincar´e inequality, we prove a metric analogue of the Bourgain-Brezis-Mironescu formula for functions in the Sobolev space W ,p ( X, d, ν ), under the assumptionthat for ν -a.e. point the tangent space in the Gromov-Hausdorff sense is Euclidean withfixed dimension N . Motivations
In this paper, we focus on the characterisation of Sobolev and BV functions in metricspaces using integrated differential quotients. Our principal motivation is the paper [4],in which the authors prove the following characterisation of Sobolev functions on opensubsets of R N . Theorem 1.1 (Bourgain, Brezis, Mironescu ’01) . Suppose that Ω ⊂ R N is a smoothbounded domain. Assume that f ∈ L p (Ω) , where p ∈ (1 , ∞ ) . Let ρ n be a sequenceof nonnegative radial mollifiers such that R R N ρ n dx = 1 and for every δ > we have lim n →∞ R ∞ δ ρ n ( r ) r N − dr = 0 . Then:(1) u ∈ W ,p (Ω) if and only if lim inf n →∞ Z Ω Z Ω | f ( x ) − f ( y ) | p | x − y | p ρ n ( x − y ) dx dy < ∞ . (2) When u ∈ W ,p (Ω) , then lim n →∞ Z Ω Z Ω | f ( x ) − f ( y ) | p | x − y | p ρ n ( x − y ) dx dy = K p,N k∇ f k pL p (Ω) . A similar result holds for p = 1 with the space BV (Ω) in place of W , (Ω), see [8].Moreover, the authors of [4] (see also [20]) prove a precompactness result under an addi-tional assumption that ρ is nonincreasing as a function of r ; this result (or a similar result Date : April 21, 2020.
Key words and phrases.
Nonlocal problems, Difference quotients, Metric measure spaces.2010
Mathematics Subject Classification: proved in [3, Theorem 6.11]) is a standard argument in approximations of local problemsvia nonlocal ones, see for instance [3, 10].A few authors, for instance [9] and [17], considered extensions of the first part of Theo-rem 1.1 to the setting of measure metric spaces. Let X be a metric space equipped witha doubling measure which satisfies the (1 , p )-Poincar´e inequality. Consider the followingmetric analogue of the left hand side of the equality in Theorem 1.1: Q r,p ( f ) = 1 r p Z X − Z B ( x,r ) | f ( x ) − f ( y ) | p dν ( y ) dν ( x ) . We will call Q r,p the BBM difference quotient. It corresponds to taking the mollifiers ρ n equal to characteristic functions of balls rescaled by the measure of these balls (see thediscussion in Section 4.1). Then, we ask if a following analogue of Theorem 1.1 holds:there exists a constant C p,X such that for any f ∈ W ,p ( X, d, ν ) we havelim r → r p Z X − Z B ( x,r ) | f ( x ) − f ( y ) | p dν ( y ) dν ( x ) = C p,X k∇ f k pL p ( X,ν ) for a gradient ∇ f understood in an appropriate sense. In such generality, there is no hopeof an exact analogue of the second part of Theorem 1.1, see Example 4.4. However, thereare some results concerning the first part of Theorem 1.1, concerning the upper and lowerlimits of Q r,p and their relationship with the Sobolev structure. Theorem 1.2. ([17, Theorem 3.1])
Let ( X, d, ν ) be a metric space equipped with a doublingmeasure which satisfies the (1 , -Poincar´e inequality. Suppose that f ∈ L ( X, ν ) . Then f ∈ BV ( X, d, ν ) ⇔ lim inf r → r Z X Z B ( x,r ) | f ( y ) − f ( x ) | p ν ( B ( x, r )) p ν ( B ( y, r )) dν ( y ) dν ( x ) < ∞ . In particular, since ν is doubling, we have f ∈ BV ( X, d, ν ) ⇔ lim inf r → r Z X − Z B ( x,r ) | f ( y ) − f ( x ) | dν ( y ) dν ( x ) < ∞ , see the discussion in [18] . The proof given in [17] with minor modifications can also be used to provide a charac-terisation of the Sobolev space W ,p ( X, d, ν ) via the lower limit of Q r,p . A similar charac-terisation for p >
1, which also arises from taking a particular kernel ρ n in Theorem 1.1and involves the limit of fractional Sobolev norms, was proved in [9].In this paper, we concentrate on the metric analogues of the second part of Theorem1.1, namely on the existence and exact value of the constant C p,X . We focus on the case p > BM APPROACH IN METRIC SPACES 3 spaces that locally look like Euclidean spaces; to be more precise, we consider spacessuch that their tangents (in the Gromov-Hausdorff sense) for ν -a.e. x ∈ X are Euclideanspaces with a fixed dimension N . This class contains for instance Riemannian manifolds,weighted Euclidean spaces for continuous weights bounded from below and (as was shownin [5]) RCD( K, N ) spaces. In absence of scaling and Taylor formula that are avalaible tous in the Euclidean case, we will use a blow-up technique and a version of the Rademachertheorem in their place.The structure of the paper is as follows. In Section 2 we recall the necessary notions,such as the (equivalent) definitions of Sobolev spaces on a metric measure space, Gromov-Hausdorff convergence and the Rademacher theorem. In Section 3, we start by provinga pointwise result (valid ν -a.e.) in the spirit of Theorem 1.1 for Lipschitz functions andthen prove the main result of the paper, Theorem 3.5: Theorem 3.5
Suppose that ( X, d, ν ) is a complete, separable, doubling metric mea-sure space which supports a (1 , p ) -Poincar´e inequality. Suppose additionally that X hasEuclidean tangents of dimension N for ν -a.e. x ∈ X . Let f ∈ W ,p ( X, d, ν ) , where p ∈ (1 , ∞ ) . Then lim r → r p Z X − Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( y ) d L N ( x ) = C p,N · Ch p ( f ) , (1.1) where Ch p ( f ) is the Cheeger energy of f defined in (2.1) and C p,N is the constant definedin (3.9) . In particular, the constant C p,X does not depend on the space X itself, only on thedimension of the tangent space, so we denote it by C p,N . Finally, in Section 4, we com-ment on the relationship of results from Section 3 with existing literature and discusssome extensions of the framework under which they are valid; in particular, we prove ananalogue of Theorem 3.5 when the tangent is the Heisenberg group and use it to constructExample 4.4 showing that if the tangent space varies from point to point, then equation(1.1) may no longer be true. 2. Preliminaries
Sobolev spaces on a metric space.
Let (
X, d, ν ) be a metric measure space. Inthe whole paper, we will work under the standard assumptions that the measure ν isdoubling and the space supports a (1 , p )-Poincar´e inequality. We say that the measure ν is doubling, if there exists a constant c D such that for all x ∈ X and all r > < ν ( B ( x, r )) ≤ c D ν ( B ( x, r )) < ∞ . W. G ´ORNY
Given f : X → R , we define its slope (also called the local Lipschitz constant of f ) bythe formula Lip( f )( x ) = lim sup y → x | f ( y ) − f ( x ) | d ( x, y ) . We say that the metric measure space (
X, d, ν ) supports a (1 , p )-Poincar´e inequality, ifthere exist constants c P and Λ such that for all f ∈ Lip( X ) and r > − Z B ( x,r ) (cid:12)(cid:12)(cid:12)(cid:12) f − (cid:18) − Z B ( x,r ) f dν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dν ≤ c P r (cid:18) − Z B ( x, Λ r ) (Lip( f )) p dν (cid:19) /p . In this paper, we will work in the setting in which the several known notions of Sobolevspaces defined on a metric space are equivalent; for completeness, we present an “H type”definition via approximation by Lipschitz functions.
Definition 2.1.
Let p ∈ (1 , ∞ ) . We say that g ∈ L p ( X, ν ) is a p -relaxed slope of f ∈ L p ( X, ν ) , if there exist e g ∈ L p ( X, ν ) and Lipschitz functions f n ∈ L p ( X, ν ) ∩ Lip ( X ) suchthat: (1) f n → f in L p ( X, ν ) and Lip ( f n ) ⇀ e g weakly in L p ( X, ν ) ; (2) e g ≤ g ν -a.e. in X .We say that g is the minimal p -relaxed slope of f if its norm in L p ( X, ν ) is minimalamong p -relaxed slopes. We will denote the minimal p -relaxed slope by |∇ f | ∗ ,p . The definition of minimal p -relaxed slope is well-posed thanks to Mazur’s lemma anduniform convexity of L p ( X, ν ), see the discussion after [2, Definition 4.2]. Using theminimal p -relaxed slope, for p ∈ (1 , ∞ ) define the Cheeger energy asCh p ( f ) = Z X |∇ f | p ∗ ,p dν. (2.1) Definition 2.2.
Fix p ∈ (1 , ∞ ) . Let f ∈ L p ( X, ν ) . We say that f ∈ W ,p ( X, d, ν ) , theSobolev space of functions with a p -relaxed slope, if there exists a p -relaxed slope of f . Thespace W ,p ( X, d, ν ) is endowed with the norm k f k W ,p ( X,d,ν ) = (cid:18) k u k pL p ( X,ν ) + Ch p ( f ) (cid:19) /p . Under the assumptions that ν is doubling and the space supports a (1 , p )-Poincar´einequality, the space W ,p ( X, d, ν ) is reflexive and bounded Lipschitz functions withbounded support form a dense subset (see [1, Corollary 7.5, Proposition 7.6]). The space W ,p ( X, d, ν ) can equivalently be defined in a few other ways: instead of the p -relaxedslope |∇ f | ∗ ,p , we may use the Cheeger’s gradient |∇ f | C,p , the p -upper gradient |∇| S,p orthe minimal p -weak upper gradient |∇| w,p ; for these equivalent definitions (all the abovegradients agree ν -a.e. in X ) and the proof of the equivalence see [2]. BM APPROACH IN METRIC SPACES 5
In the proofs in Section 3, we are going to use one more equivalence of Sobolevspaces - with the Hajlasz-Sobolev space M ,p ( X ) (see Lemma 2.3). While the normsin W ,p ( X, d, ν ) and M ,p ( X ) do not necessarily agree, classical arguments using maximalfunctions (for instance, combine [15, Theorem 4.5] and [14, Theorem 1.0.1]) imply thefollowing Lemma concerning the equivalence of these spaces. Lemma 2.3.
Let p ∈ (1 , ∞ ) . Suppose that ( X, d, ν ) is a doubling metric measure spacewhich supports a (1 , p ) -Poincar´e inequality. Then, for any f ∈ W ,p ( X, d, ν ) there exists g ∈ L p ( X, ν ) such that | f ( x ) − f ( y ) | ≤ d ( x, y ) ( g ( x ) + g ( y )) for ν -a.e. x, y ∈ X ( in other words, f is in the Hajlasz-Sobolev space M ,p ( X )) . Moreover,we can choose g such that k g k pL p ( X,ν ) ≤ C · Ch p ( f ) . Tangents of a metric space.
Let us recall the definition of pointed measuredGromov-Hausdorff convergence of metric spaces (first introduced in [11]; there are manyequivalent ways to define it in the literature, we use a variant from [7]).
Definition 2.4.
A map φ : ( X , x , d ) → ( X , x , d ) between two metric spaces with adistinguished point is called an ε -isometry if | d ( φ ( x ) , φ ( y )) − d ( x, y ) | ≤ ε for all x, y ∈ B ( x, ε − ) and we have B d ( y, r − ε ) ⊂ N ε ( φ ( B d ( x, r ))) for all r ∈ [ ε − , ε ] . Here, N ε ( E ) denotes the open ε -neighbourhood of a set E ⊂ X . In particular, we do not necessarily have that φ ( x ) = x , but the properties of an ε -isometry imply that d ( φ ( x ) , x ) ≤ ε . Definition 2.5.
A sequence of pointed metric spaces ( X n , x n , d n ) converges in pointedGromov-Hausdorff sense to ( X, x, d ) if there exists a sequence ε n → such that thereexist ε n -isometries φ n : X n → X and ψ n : X → X n .Moreover, we say that ( X n , x n , d n , ν n ) converges in measured pointed Gromov-Hausdorffsense to ( X, x, d, ν ) , if additionally ( φ n ) ν n ⇀ ν weakly as measures on X . Definition 2.6.
Let ( X, x, d, ν ) be a pointed metric measure space. A tangent cone at x isa pointed metric space ( X ∞ , x ∞ , d ∞ , ν ∞ ) , which is a measured pointed Gromov-Hausdorfflimit of some sequence ( X, x, r − n d, ν r n ) , where r n → and ν r = 1 ν ( B ( x, r )) ν. W. G ´ORNY
In the literature the renormalised limit measure µ ∞ is sometimes omitted in the definitionof tangent cones; here, we follow [6] and include it, since we want to use a version ofRademacher’s theorem. On complete metric spaces equipped with a doubling measure tangent cones exist forall x ∈ X , see [6], but they are not necessarily unique. A key assumption we will useis that for ν -a.e. x ∈ X the tangent cones are unique and are Euclidean spaces of fixeddimension N . In this case, we will drop the sequence r n and simply index the blow-upsof the space X by r ∈ (0 , ∞ ).2.3. Rademacher theorem.
The core of the proofs in the next Section is a version ofthe Rademacher theorem for metric measure spaces which satisfy the doubling propertyand the (1 , p )-Poincar´e inequality. To this end, we introduce the following notation.
Notation.
Set φ r : X → X ∞ to be the Gromov-Hausdorff approximation. Given afunction f ∈ Lip( X ), we denote f r,x ( y ) = f ( y ) − f ( x ) r . We have f r,x ( y ) ∈ Lip( X ); moreover, if L is the Lipschitz constant of f , then the Lipschitzconstant of f r,x is at most Lr and | f r,x | is bounded by L on the ball B ( x, r ). If we rescalethe metric d to r − d , then f r,x has Lipschitz constant at most L , is locally bounded and isbounded by L on the ball with radius one; hence, it admits a convergent subsequence (stilldenoted by f r,x ) such that f r,x converge locally uniformly to a function f ,x ∈ Lip( X ∞ )(modulo the identification of X as a subset of X ∞ via φ r ), namely on B ( x, r ) we have k f ,x ( φ r ( · )) − f r,x ( · ) k ∞ ≤ α ( r ) , (2.2)where α ( r ) → r →
0. Moreover, the Lipschitz constant of f ,x is at most L and it isbounded by L on the ball B ( x ∞ , g f the minimal upper gradient of a function f ∈ W ,p ( X, d, ν ). Definition 2.7.
Let p ∈ (1 , ∞ ) . A Lipschitz function l ∈ Lip ( X ) is generalised linear if:(1) l ≡ or range l = ( −∞ , ∞ ) ;(2) l is p-harmonic, in the sense that for any V ⊂⊂ X we have Z V | g l | p ≤ Z V | g l + f | p for all functions f ∈ W ,p ( X, d, ν ) with support in V ;(3) g l ≡ c for some c ∈ R . BM APPROACH IN METRIC SPACES 7 If X is the Euclidean space, then generalised linear functions are affine, see [6, Theorem8.11]. Theorem 2.8. ([6, Theorem 10.2])
Suppose that ( X, x, d, ν ) is a pointed metric measurespace. Suppose that ν is doubling and satisfies the (1 , p ) -Poincar´e inequality for some p ∈ (1 , ∞ ) . Let f ∈ Lip ( X ) Then, for ν -a.e. x ∈ X the function f is infinitesimallygeneralised linear, i.e. for all p ′ > p any f ,x as above is a generalised linear function.Moreover, we have Lip f ,x = Lip ( f )( x ) . Bourgain-Brezis-Mironescu approach
In this Section, we deal with metric measure spaces (
X, d, ν ) which have Euclideantangents ν -a.e., i.e.( X, x, r − d, ν r ) → ( X ∞ , x ∞ , d ∞ , ν ∞ ) = ( R N , , k · k , c N L N )in the measured Gromov-Hausdorff sense, where constant c N = L N ( B (0 , , so that themeasure of the unit ball equals one (this is a consequence of the definition of ν r ). This isthe case for instance for Riemannian manifolds and (as shown in [5]) RCD( K, N ) spaces.Another important class of examples are weighted Euclidean spaces.
Example 3.1.
Let ( X, x, d, ν ) = ( R N , x, k · k , w L N ) , where w ∈ L loc ( R N ) is continuous L N -a.e. and L N -a.e. we have w ≥ c > . Choose x ∈ X which satisfies these conditionsand define φ r : ( R N , x, r − k · k ) → ( R N , , k · k ) by the formula φ r ( y ) = y − xr and noticethat it is an isometry (with an inverse which is also an isometry) which maps x to ,so the spaces ( R N , x, r − k · k ) converge in the Gromov-Hausdorff sense to ( R N , , k · k ) .Moreover, a quick calculation shows that ( φ r ) ν r ( z ) = r N w ( x + rz ) R B ( x,r ) w d L N d L N ( z ) = c N w ( x + rz ) − R B ( x,r ) w d L N d L N ( z ) ⇀ c N L N . Hence, ( R N , , k · k , c N L N ) satisfies all the conditions given in Definition 2.6 for anysubsequence r n → , so ( X, x, d, ν ) has Euclidean tangents ν -a.e. The goal of this Section is to prove Theorem 3.5, which is an equivalent of Theorem 1.1in the metric setting.The outline of the proof is in a way similar to the proof of Theorem1.1 shown in [4]: first, we prove a pointwise result for a dense subset of the Sobolev spacewhich contains functions which are regular enough, and then integrate this result over thewhole space and prove that the limiting process is well defined. Here, we further breakthis reasoning into separate results in order to underline the moment when we use theassumption that the tangent spaces are Euclidean.
W. G ´ORNY
Lemma 3.2.
Suppose that ( X, d, ν ) is a doubling metric measure space, which satisfiesthe (1 , p ) -Poincar´e inequality for some p ∈ (1 , ∞ ) . Let x ∈ X be a point such thatthe implication in the Rademacher theorem (Theorem 2.8) holds. Then, in the notationintroduced in Section 2.2, we have lim r → (cid:18) Z B ( x,r ) | f r,x ( y ) | p dν r ( y ) − Z B ( x ∞ , | f ,x ( z ) | p d ( φ r ) ν r ( z ) (cid:19) = 0 . (3.1)This result will later play a role as an estimate on the remainder, when we will ap-proximate the rescaled nonlocal gradients f x,r by the linear part f ,x . Compared to thesituation when X = R N , the main difference is that there are two sources of error here -one which is of the same type as the Taylor remainder and one that comes from the factthat the domain changes in the approximation; it reflects the difference in the shapes ofballs B ( x, r ) and the ball B ( x ∞ , Proof.
Fix such x ∈ X such that the Rademacher theorem holds (the set of such points isof full measure). Take the functions f r,x , which by Arzela-Ascoli theorem converge locallyuniformly (on a subsequence still denoted by r ) to a function f ,x . As discussed in Section2.2, on B ( x, r ) we have k f ,x ( φ r ( · )) − f r,x ( · ) k ∞ ≤ α ( r ) , where α ( r ) → r →
0. Now, write the left integral in (3.1) as Z B ( x,r ) | f r,x ( y ) | p dν r ( y ) = Z B ( x,r ) | f ,x ( φ r ( y )) | p dν r ( y )+ (3.2)+ Z B ( x,r ) (cid:18) | f r,x ( y ) | p − | f ,x ( φ r ( y )) | p (cid:19) dν r ( y ) . We start by estimating the second summand on the right hand side. By the Lagrangemean value theorem for φ ( t ) = t p we have that for any y ∈ B ( x, r ) (cid:12)(cid:12)(cid:12)(cid:12) | f r,x ( y ) | p − | f ,x ( φ r ( y )) | p (cid:12)(cid:12)(cid:12)(cid:12) = p τ p − | f r,x ( y ) − f ,x ( φ r ( y )) | for some τ between | f r,x ( y ) | and | f ,x ( φ r ( y )) | . But by definition of f r,x we have that | f r,x | is bounded by Lip( f )( x ) on the ball B ( x, r ); since f ,x is the uniform limit of f r,x as r → B ( x, r ), it satisfies the same bound. Hence, taking (2.2) into account, we have that p τ p − | f r,x ( y ) − f ,x ( φ r ( y )) | ≤ p | Lip( f )( x ) | p α ( r )for all y ∈ B ( x, r ). Coming back to (3.2), we have (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x,r ) (cid:18) | f r,x ( y ) | p − | f ,x ( φ r ( y )) | p (cid:19) dν r ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B ( x,r ) (cid:12)(cid:12)(cid:12)(cid:12) | f r,x ( y ) | p − | f ,x ( φ r ( y )) | p (cid:12)(cid:12)(cid:12)(cid:12) dν r ( y ) = BM APPROACH IN METRIC SPACES 9 = − Z B ( x,r ) (cid:12)(cid:12)(cid:12)(cid:12) | f r,x ( y ) | p − | f ,x ( φ r ( y )) | p (cid:12)(cid:12)(cid:12)(cid:12) dν ( y ) ≤ p | Lip( f )( x ) | p α ( r ) , so lim r → Z B ( x,r ) (cid:18) | f r,x ( y ) | p − | f ,x ( φ r ( y )) | p (cid:19) dν r ( y ) = 0 . (3.3)To finish the proof, we need to show that the expression Z B ( x,r ) | f ,x ( φ r ( y )) | p dν r ( y ) − Z B ( x ∞ , | f ,x ( z ) | p d ( φ r ) ν r ( z )goes to zero as r →
0. Notice that Z B ( x,r ) | f ,x ( φ r ( y )) | p dν r ( y ) = Z φ r ( B ( x,r )) | f ,x | p d ( φ r ) ν r = Z B ( x ∞ , | f ,x | p d ( φ r ) ν r ++ Z φ r ( B ( x,r )) \ B ( x ∞ , | f ,x | p d ( φ r ) ν r − Z B ( x ∞ , \ φ r ( B ( x,r )) | f ,x | p d ( φ r ) ν r , (3.4)so we have to prove that the second and third summand on the right hand side of (3.4)disappear in the limit r → φ r are ε r -isometries. For any x, y ∈ X we have (cid:12)(cid:12)(cid:12)(cid:12) d ∞ ( φ r ( x ) , φ r ( y )) − r − d ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε r , (3.5)so for y ∈ B ( x, r ) we have d ∞ ( φ r ( y ) , x ∞ ) ≤ r − d ( x, y ) + ε r + d ∞ ( φ r ( x ) , x ∞ ) ≤ ε r . In other words, φ r ( B ( x, r )) \ B ( x ∞ , ⊂ B ( x ∞ , ε r ) \ B ( x ∞ , ε r → r → ρ k small enough that ε r < k for all r ∈ (0 , ρ k ]. On the ball B ( x ∞ , B ( x ∞ , ε r ) \ B ( x ∞ ,
1) for r ∈ (0 , ρ k ), the function | f ,x | is uniformly boundedby some M , solim sup r → (cid:12)(cid:12)(cid:12)(cid:12) Z φ r ( B ( x,r )) \ B ( x ∞ , | f ,x | p d ( φ r ) ν r (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup r → M p (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x ∞ , ε r ) \ B ( x ∞ , d ( φ r ) ν r (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ lim sup r → M p (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x ∞ , k ) \ B ( x ∞ , d ( φ r ) ν r (cid:12)(cid:12)(cid:12)(cid:12) = M p ν ∞ ( B ( x ∞ , k ) \ B ( x ∞ , . Recall that doubling measures (and ν ∞ is doubling as a limit of a uniformly doublingsequence) give zero measure to boundaries of balls, so we have ν ∞ ( ∂B ( x ∞ , k was arbitrary, the right hand side can be made arbitrarily small and we see thatlim r → (cid:18) Z φ r ( B ( x,r )) \ B ( x ∞ , | f ,x | p d ( φ r ) ν r (cid:19) = 0 . (3.6) We estimate the third summand in the right hand side of (3.4) as follows. For any x ∈ X and y / ∈ B ( x, r ), by (3.5) we have d ∞ ( φ r ( y ) , x ∞ ) ≥ r − d ( x, y ) − ε r − d ∞ ( φ r ( x ) , x ∞ ) k ≥ − ε r . Again, fix ρ k small enough that ε r < k for all r ∈ (0 , ρ k ]; then the inequality above meansthat φ r ( X ) \ φ r ( B ( x, r )) ⊂ X ∞ \ B ( x ∞ , − ε r ) ⊂ X ∞ \ B ( x ∞ , − k ). By definition of apushforward measure, ( φ r ) ν r is supported on the image of φ r , solim sup r → (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x ∞ , \ φ r ( B ( x,r )) | f ,x | p d ( φ r ) ν r (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup r → M p (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x ∞ , \ φ r ( B ( x,r )) d ( φ r ) ν r (cid:12)(cid:12)(cid:12)(cid:12) == lim sup r → M p (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x ∞ , ∩ ( φ r ( X ) \ φ r ( B ( x,r ))) d ( φ r ) ν r (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ lim sup r → M p (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x ∞ , \ B ( x ∞ , − k ) d ( φ r ) ν r (cid:12)(cid:12)(cid:12)(cid:12) = M p ν ∞ ( B ( x ∞ , \ B ( x ∞ , − k )) . Since k was arbitrary, the right hand side can be made arbitrarily small and we see thatlim r → (cid:18) Z B ( x ∞ , \ φ r ( B ( x,r )) | f ,x | p d ( φ r ) ν r (cid:19) = 0 . (3.7)When we plug in equations (3.6) and (3.7) to (3.4), we obtain thatlim r → (cid:18) Z B ( x,r ) | f ,x ( φ r ( y )) | p dν r ( y ) − Z B ( x ∞ , | f ,x ( z ) | p d ( φ r ) ν r ( z ) (cid:19) = 0 , which together with (3.3) give the statement of the Lemma. ✷ Proposition 3.3.
Suppose that ( X, d, ν ) is a doubling metric measure space, which sat-isfies the (1 , p ) -Poincar´e inequality for some p ∈ (1 , ∞ ) . Suppose additionally that X has ν -a.e. Euclidean tangents of dimension N . Let f ∈ Lip ( X ) . Then for ν -a.e. x ∈ X wehave lim r → r p − Z B ( x,r ) | f ( x ) − f ( y ) | p dν ( y ) = C p,N | Lip ( f )( x ) | p , (3.8) where C p,N = − Z B (0 , | z · v | p d L N ( z ) , (3.9) where v is any unit vector in R N . This constant is not the same as the constant K p,N inthe statement of Theorem 1.1, but they are closely related, see Section 4.1. BM APPROACH IN METRIC SPACES 11
Proof.
The set of points in the statement of Theorem 2.8 (Rademacher theorem) is of fullmeasure; choose such a point x ∈ X . In the notation introduced in Section 2.2, noticethat | f ( x ) − f ( y ) | = r | f r,x ( y ) | , so we may use Lemma 3.2 in the last equality and obtainlim r → r p − Z B ( x,r ) | f ( x ) − f ( y ) | p dν ( y ) = lim r → r p Z B ( x,r ) | f ( x ) − f ( y ) | p dν r ( y ) == lim r → Z B ( x,r ) | f r,x ( y ) | p dν r ( y ) = lim r → Z B ( x ∞ , | f ,x ( z ) | p d ( φ r ) ν r ( z ) . Now, we need to estimate this last expression using the fact that for ν -a.e. x ∈ X thetangent space is the Euclidean space ( R N , , k · k , c N L N ).Recall that f ,x is a generalised linear function with Lipschitz constant Lip( f )( x ). Sincethe tangent space X ∞ is Euclidean, by [6, Theorem 8.11] f ,x is affine; since f ,x is thelocally uniform limit of f r,x , it has value 0 at zero. This means that f ,x is of the form f ,x ( z ) = Lip( f )( x ) z · v, where v is a vector of length one. Since (by definition of measured Gromov-Hausdorffconvergence) the measures ( φ r ) ν r converge weakly to c N L N , we havelim r → (cid:18) Z B ( x ∞ , | f ,x ( z ) | p d ( φ r ) ν r ( z ) (cid:19) = lim r → (cid:18) | Lip( f )( x ) | p Z B (0 , | z · v | p d ( φ r ) ν r (cid:19) == (cid:18) Z B (0 , | z · v | p c N d L N ( z ) (cid:19) | Lip( f )( x ) | p = C p,N | Lip( f )( x ) | p , where C p,N is the constant introduced in (3.9); note that it only depends on p and thedimension of the tangent space. ✷ This approach, using a blow-up technique and the Rademacher theorem instead of theTaylor formula used in the original proof in [4], gives a new proof even in the contextof Euclidean spaces. Moreover, a significant part of the proof did not depend on thestructure of the tangent space; it plays a role only via the characterisation of generalisedlinear functions. Therefore, this approach allows for some extensions in terms of thestructure of the tangent space, such as the case when the tangent space at ν -a.e. pointis a fixed Carnot group G of step 2, see Section 4.2. Finally, notice that the constant C p,N does not depend on the metric space itself - it depends only on the dimension of thetangent space N .Now, we use the pointwise result proved above to prove the desired result for Sobolevspaces for p >
1. The first step is to prove a uniform estimate on the integral of thenonlocal gradient for Sobolev functions. From now on, denote ∆ r = { ( x, y ) ∈ X × X : d ( x, y ) < r } . Lemma 3.4.
Let p ∈ (1 , ∞ ) . Suppose that ( X, d, ν ) is a doubling metric measure spacewhich supports a (1 , p ) -Poincar´e inequality. For any f ∈ W ,p ( X, d, ν ) we have r p Z X − Z B ( x,r ) | f ( y ) − f ( x ) | p dν ( y ) dν ( x ) ≤ C ( p, X ) · Ch p ( f ) . (3.10) Proof.
Take g ∈ L p ( X, ν ) given by Lemma 2.3 and calculate1 r p Z X − Z B ( x,r ) | f ( y ) − f ( x ) | p dν ( y ) dν ( x ) = 1 ν ( B ( x, r )) Z ∆ r (cid:12)(cid:12)(cid:12)(cid:12) f ( y ) − f ( x ) r (cid:12)(cid:12)(cid:12)(cid:12) p dν ( y ) dν ( x ) ≤≤ ν ( B ( x, r )) Z ∆ r (cid:12)(cid:12)(cid:12)(cid:12) f ( y ) − f ( x ) d ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) p dν ( y ) dν ( x ) ≤ C p ν ( B ( x, r )) Z ∆ r ( g ( x ) + g ( y )) p dν ( y ) dν ( x ) ≤≤ C p p − ν ( B ( x, r )) Z ∆ r (cid:18) ( g ( x )) p + ( g ( y )) p (cid:19) dν ( y ) dν ( x ) == C p p − ν ( B ( x, r )) (cid:18) Z X Z B ( x,r ) ( g ( x )) p dν ( y ) dν ( x ) + Z B ( y,r ) Z X ( g ( y )) p dν ( y ) dν ( x ) (cid:19) == C p p − Z X ( g ( x )) p dν ( x ) + C p p − ν ( B ( y, r )) ν ( B ( x, r )) Z X ( g ( y )) p dν ( y ) ≤≤ C ′ Z X ( g ( x )) p dν ( x ) ≤ C ( p, X ) · Ch p ( f ) . Here, the constant in the last line comes from Lemma 2.3 and the doubling property. ✷ Now, we integrate the pointwise result (Proposition 3.3) and use the density of Lipschitzfunctions to prove an analogue of Theorem 1.1 for Sobolev spaces W ,p ( X, d, ν ) for p >
Theorem 3.5.
Suppose that ( X, d, ν ) is a complete, separable, doubling metric mea-sure space which supports a (1 , p ) -Poincar´e inequality. Suppose additionally that X hasEuclidean tangents of dimension N for ν -a.e. x ∈ X . Let f ∈ W ,p ( X, d, ν ) , where p ∈ (1 , ∞ ) . Then lim r → r p Z X − Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( y ) d L N ( x ) = C p,N · Ch p ( f ) . (3.11) Proof.
Set f r ( x, y ) = | f ( x ) − f ( y ) | r χ B ( x,r ) ( y ) | B ( x, r ) | − /p ∈ L p ( X × X, ν ⊗ ν ) . Using this function, we can rephrase equation (3.11) aslim r → k f r k pL p ( X × X,ν ⊗ ν ) = C p,N · Ch p ( f ) BM APPROACH IN METRIC SPACES 13 and equation (3.10) as k f r k pL p ( X × X,ν ⊗ ν ) ≤ C · Ch p ( f ). Now, take any f, g ∈ W ,p ( X, d, ν ).We estimate (cid:12)(cid:12)(cid:12)(cid:12) k f r k L p ( X × X,ν ⊗ ν ) − k g r k L p ( X × X,ν ⊗ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ p − k ( f − g ) r k L p ( X × X,ν ⊗ ν ) ≤ C · (Ch p ( f − g )) /p . By the above equation, taking into account the density of bounded Lipschitz functionswith bounded support in W ,p ( X, d, ν ), it suffices to establish equation (3.11) for Lip( X ).Take any f ∈ Lip( X ) with Lipschitz constant L and use Proposition 3.3; for ν − a.e. x ∈ X we obtain equality (3.8). Then1 r p − Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( x ) ≤ − Z B ( x,r ) L p d L N ( x ) = L p . Hence, we may integrate equality (3.8) over X and use the dominated convergence theoremto change the order of integration and taking the limit; we get that (3.11) is satisfied for f . We extend this result to W ,p ( X, d, ν ) by density of Lipschitz functions. ✷ Comments and extensions
Comparison with taking averages on balls.
The constant K p,N is Theorem 1.1and the constant C p,N in Theorem 3.5 are not equal, but they are closely related; in thecase when X = R N , the two results are related as follows: if we make the right choice ofthe approximating kernel ρ r in Theorem 1.1, namely ρ r ( x ) = (cid:18) r N Z B (0 , | z | p d L N ( z ) (cid:19) − | x | p r p χ B (0 ,r ) ( x ) , we get Theorem 3.5. Such ρ r satisfies the assumptions of Theorem 1.1, since it is nonneg-ative, radial, has support in the ball B (0 , r ) and the normalisation constant is chosen sothat R R N ρ r d L N = 1. If we use such ρ r in Theorem 1.1, we obtain K p,N k∇ f k pL p ( R N ) = lim r → Z R N Z R N | f ( x ) − f ( y ) | p | x − y | p ρ r ( | x − y | ) d L N ( x ) d L N ( y ) == lim r → (cid:18) r N Z B (0 , | z | p d L N ( z ) (cid:19) − r p Z R N Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( x ) d L N ( y ) == lim r → (cid:18) − Z B (0 , | z | p d L N ( z ) (cid:19) − r p Z R N − Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( x ) d L N ( y ) , solim r → r p Z R N − Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( x ) d L N ( y ) = (cid:18) − Z B (0 , | z | p d L N ( z ) (cid:19) K p,N k∇ f k pL p ( R N ) . Hence, we have C p,N = (cid:18) − Z B (0 , | z | p d L N ( z ) (cid:19) K p,N . Finally, let us see that it agrees with the value given in Proposition 3.3. We use thespherical version of the Fubini theorem with u ( x ) = χ B (0 , ( x ) | x | p : C p,N = K p,N L N ( B (0 , Z B (0 , | z | p d L N ( z ) = K p,N L N ( B (0 , Z r N + p − Z ∂B (0 , dσ dr == H N − ( S N − ) K p,N L N ( B (0 , Z r N + p − dr = H N − ( S N − ) L N ( B (0 , Z r N + p − − Z ∂B (0 , | x · v | p dσ dr == 1 L N ( B (0 , Z r N − Z ∂B (0 , | rx · v | p dσ dr == 1 L N ( B (0 , Z ∞ r N − Z ∂B (0 , χ B (0 , ( rx ) | rx · v | p dσ dr == 1 L N ( B (0 , Z R N χ B (0 , ( x ) | x · v | p d L N ( x ) = − Z B (0 , | x · v | p d L N ( x ) , hence, the constant C p,N is consistent with the constant K p,N for a special choice of theapproximating sequence.4.2. Spaces with Heisenberg group as a tangent.
A closer look at the structureof the proof of Theorem 3.5 reveals that the assumption that X has Euclidean tangents ν -a.e. comes into play only via the structure of generalized linear functions on the tangentspace X ∞ . Therefore, in principle it should be possible to generalize Theorem 3.5 to thecase when the tangent space at ν -a.e. point is fixed, but not Euclidean. In this Section,we take a closer look at the classical results of Cheeger ([6]) to present such an argumentfor a simple case: the Heisenberg group H .Recall that the Heisenberg group H is the space R equipped with a Lie group structurewith multiplication( x , x , x ) · ( y , y , y ) = ( x + y , x + y , x + y + 2( x y − x y ))and equipped with the Carnot-Carath´eodory distance (arising from a family of left in-variant vector fields). By the left invariance of the distance, it is enough to compute thedistance from 0 to any given point (denoted by d ); then, the distance d H is related to BM APPROACH IN METRIC SPACES 15 d by left invariance, namely d H ( x, y ) = d ( y − x ). As proved in [12, Corollary 3.2], d isgiven by the formula d (cid:18) ( x , x , x ) (cid:19) = x p x + x sin (cid:18) πH − ( x x + x ) (cid:19) + q x + x cos (cid:18) πH − ( x x + x ) (cid:19) , (4.1)where H : ( − , → R is defined by the formula H ( s ) = 2 π − cos(2 πs ) (cid:18) s − sin(2 πs )2 π (cid:19) . The function H is a real analytic diffeomorphism of ( − ,
1) onto R with H (0) = 0.We begin the argument by recalling [6, Theorem 8.10]. Theorem 4.1.
Assume that Z is complete, noncompact, equipped with a doubling measure µ which satisfies the (1 , p ) -Poincar´e inequality. Let l ∈ Lip ( Z ) be a generalised linearfunction on Z . Then, for any z ∈ Z there exists a geodesic γ : ( −∞ , ∞ ) → Z with γ (0) = z such that γ is an integral curve for the upper gradient g l = Lip ( l ) . Next, we set b γ,s ( z ) = d ( z, γ ( s )) − | s | , and define the Busemann functions b ± γ ( z ) by theformula b ± γ ( z ) = lim s →±∞ b γ,s ( z ). The limit is well defined since the b γ,s is decreasing in | s | and bounded from below on compact subsets of Z . Now, we recall [6, Theorem 8.11]. Theorem 4.2.
Under the assumptions of Theorem 4.1, for any geodesic γ as given bythat Theorem, we have l ( z ) − Lip ( l ) · b + γ ( z ) ≤ l ( z ) ≤ l ( z ) + Lip ( l ) · b − γ ( z ) . (4.2)Our goal is to analyse the Busemann functions to show that on the Heisenberg group H these inequalities are in fact equalities (as in the Euclidean case), which will give astructure result on the generalised linear functions. In the case interesting to us, when Z = H , unbounded geodesics are horizontal lines (which is not true in general even forCarnot groups), see [16, Proposition 5.6]. Let γ be given by Theorem 4.1; then, since it isa horizontal line, it is of the form γ ( s ) = ( as, bs, a + b = 1. Since by equation(4.1) the distance d H is invariant with respect to rotations in the horizontal plane, withoutloss of generality we may assume that ( a, b ) = (1 , z = ( z , z , z ), we have b γ,s ( z ) = d H ( z, γ ( s )) − | s | = d (0 , ( − γ ( s )) · z ) − | s | = d (( z − s, z , z − z s )) − | s | == ( z − z s ) p ( z − s ) + z sin (cid:18) πH − ( z − z s ( z − s ) + z ) (cid:19) ++ q ( z − s ) + z cos (cid:18) πH − ( z − z s ( z − s ) + z ) (cid:19) − | s | . We will compute the limit of b γ,s as s → + ∞ ; the other calculation is similar. Recall that H ′ (0) = 0 and H (0) = 0, so on the first part we havelim s →∞ ( z − z s ) p ( z − s ) + z sin (cid:18) πH − ( z − z s ( z − s ) + z ) (cid:19) == lim s →∞ ( − z ) sin (cid:18) πH − ( z − z s ( z − s ) + z ) (cid:19) = 0 . On the second part, we havelim s →∞ (cid:18)q ( z − s ) + z cos (cid:18) πH − ( z − z s ( z − s ) + z ) (cid:19) − s (cid:19) == lim s →∞ (( z − s ) + z ) cos ( πH − ( z − z s ( z − s ) + z )) − s p ( z − s ) + z cos( πH − ( z − z s ( z − s ) + z )) + s == lim s →∞ s (cid:18) (( z − s ) + z ) cos ( πH − ( z − z s ( z − s ) + z )) − s (cid:19) == lim s →∞ s (cid:18) ( s − z s ) cos ( πH − ( z − z s ( z − s ) + z )) − s (cid:19) == lim s →∞ (cid:18) s (cos ( πH − ( z − z s ( z − s ) + z )) − − z cos ( πH − ( z − z s ( z − s ) + z )) (cid:19) = − z . Hence, we have that b + γ (( z , z , z )) = − z ; similarly, we have b − γ (( z , z , z )) = z . Inparticular, b + γ = − b − γ , so we have equalities in equation (4.2). Assuming additionallythat l ((0 , , l is of the form l (( z , z , z )) = Lip( l ) z .In general, for any horizontal line ( as, bs, v = ( a, b,
0) we have l ( z ) = Lip( l )( z · v ), where · denotes the usual scalar product in R .Now, we investigate the proof of Theorem 3.5: the only place where the assumptionthat the tangent is the Euclidean space comes into play is in the final step of the proof ofProposition 3.3. In that step, we instead proceed as follows. By the considerations above f ,x (in the notation of Proposition 3.3) is of the form f ,x ( z ) = Lip( f )( x ) z · v, where v = ( a, b,
0) with a + b = 1. By definition of measured Gromov-Hausdorff con-vergence the measures ( φ r ) ν r converge weakly to c H L , where c H = ( L ( B H (0 , − ,solim r → (cid:18) Z B ( x ∞ , | f ,x ( z ) | p d ( φ r ) ν r ( z ) (cid:19) = lim r → (cid:18) | Lip( f )( x ) | p Z B H (0 , | z · v | p d ( φ r ) ν r (cid:19) == (cid:18) Z B H (0 , | z · v | p c H d L N ( z ) (cid:19) | Lip( f )( x ) | p = C p, H | Lip( f )( x ) | p , BM APPROACH IN METRIC SPACES 17 where C p, H = − Z B H (0 , | z · v | p . Here, v is any unit horizontal vector. Note that this does not depend on the choice of v due to the invariance of the distance d H (0 , x ) with respect to horizontal rotations - it isa constant that again only depends on p and the choice of the tangent space. Therefore,we proved that Corollary 4.3.
Suppose that ( X, d, ν ) is a complete, separable, doubling metric measurespace which supports a (1 , p ) -Poincar´e inequality. Suppose additionally that the tangentspace to X for ν -a.e. x ∈ X is the Heisenberg group H . Let f ∈ W ,p ( X, d, ν ) , where p ∈ (1 , ∞ ) . Then lim r → r p Z X − Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( y ) d L N ( x ) = C p, H · Ch p ( f ) . Finally, notice that since the formula (4.1) for the distance holds also in higher Heisen-berg groups H N (as proved in [12]), the same proof works also in that case; however, forsimplicity we presented the proof for H .4.3. Spaces with tangent changing from point to point.
In this subsection, wewant to illustrate that the assumption that the tangent space is fixed is crucial in orderfor Theorem 3.5 to hold. To this end, we will use the space constructed in [13, Remark6.19(a)] by gluing together the Euclidean space R and the Heisenberg group.Suppose that A is a closed subset of a metric space Y such that an isometric copy of A lies inside a metric space Z , i.e. there exists an isometric embedding i : A → Z . Weunderstand this embedding to be fixed and consider A to be a closed subset of both Y and Z . We define the space Y ∪ A Z to be the disjoint union of Y and Z with points inthe two copies of A identified. This space is endowed with a natural metric which extendsthe original metrics in Y and Z ; given y, z ∈ Y ∪ A Z , we set d ( y, z ) = inf a ∈ A d Y ( y, a ) + d Z ( a, z ) . Example 4.4.
Let X = R ∪ A H , where A is an unbounded geodesic ( any line in R anda horizontal line in H ) . As shown in [13, Remark 6.19(a)] , this space is doubling ( even -regular ) and admits a (1 , p ) -Poincar´e inequality for all p > .Now, we take two functions with supports away from A . Namely, we set f ∈ C ∞ c ( R \ A ) and g ∈ C ∞ c ( H \ A ) . We extend them by zero to the whole space X . Then, since thesupport of f lies entirely in R , by Theorem 3.5 we have lim r → r p Z X − Z B ( x,r ) | f ( x ) − f ( y ) | p d L N ( y ) d L N ( x ) = C p, · Ch p ( f ) and since the support of g lies entirely in H , by Corollary 4.3 we have lim r → r p Z X − Z B ( x,r ) | g ( x ) − g ( y ) | p d L N ( y ) d L N ( x ) = C p, H · Ch p ( g ) . In particular, there is no single constant C p,X such that the statement of Theorem 3.5holds, since there exists p > such that C p, = C p, H ; for instance, for p = 4 we have C , = − Z B R (0 , | z · v | d L ( z ) = − Z B R (0 , | z · e | d L ( z ) = 1 π Z B R (0 , | z | d L ( z ) == 2 π Z − | z | (cid:18) Z B (( z , , , , √ − z ) d L (( z , z , z )) (cid:19) d L ( z ) == 2 π Z − | z | π (1 − z ) d L ( z ) = 83 π Z − | z | (1 − z ) d L ( z ) = 116 = 0 . , while the constant C , H ( which we can compute numerically from the explicit parametri-sation of the unit ball in H given in [19]) has value C , H ≈ . . Hence, for p = 4 the space is doubling and satisfies a (1 , p ) -Poincar´e inequality, but since it has differenttangents at different points, an analogue of Theorem 3.5 does not hold in this setting. Acknowledgements.
This work has been partially supported by the research projectno. 2017/27/N/ST1/02418 funded by the National Science Centre, Poland. The motiva-tion for writing this paper originated during my visit to the Scuola Normale Superiore diPisa; I wish to thank them for their hospitality and Luigi Ambrosio for his support.
References [1]
L. Ambrosio, M. Colombo, and S. di Marino , Sobolev spaces in metric measure spaces:reflexivity and lower semicontinuity of slope, in Advanced Studies in Pure Mathematics: Varia-tional methods for evolving objects, L. Ambrosio, Y. Giga, P. Rybka, and Y. Tonegawa, eds., Tokyo,2015, Mathematical Society of Japan, pp. 1–58.[2]
L. Ambrosio, N. Gigli, and G. Savar´e , Density of Lipschitz functions and equivalence of weakgradients in metric measure spaces, Rev. Mat. Iberoam., 29 (2013), pp. 969–996.[3]
F. Andreu-Vaillo, J. Maz´on, J. Rossi, and J. Toledo , Nonlocal Diffusion Problems, Mathe-matical Surveys and Monographs, vol. 165, AMS, 2010.[4]
J. Bourgain, H. Brezis, and P. Mironescu , Another look at Sobolev spaces, in Optimal Controland Partial Diferential Equations, J. L. M. et al., ed., Amsterdam, 2001, IOS Press, pp. 439–455.[5]
E. Bru´e and D. Semola , Constancy of the dimension for RCD(
K, N ) spaces via regularity ofLagrangian flows, Comm. Pure Appl. Math., https://doi.org/10.1002/cpa.21849, (2019).[6]
J. Cheeger , Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal.,9 (1999), pp. 428–517.[7]
G. David , Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces, Geom.Funct. Anal., 25 (2015), pp. 553–579.
BM APPROACH IN METRIC SPACES 19 [8]
J. D´avila , On an open question about functions of bounded variation, Calc. Var. Partial DifferentialEquations, 15 (2002), pp. 519–527.[9]
S. Di Marino and M. Squassina , New characterizations of Sobolev metric spaces, J. Funct. Anal.,276 (2019), pp. 1853–1874.[10]
W. G´orny , Local and nonlocal 1-Laplacian in Carnot groups, arXiv:2001.02202, (2020).[11]
M. Gromov, J. Lafontaine, and P. Pansu , Structures m´etriques pour les vari´eti´esriemanniennes, Cedic/Fernand Nathan, Paris, 1981.[12]
P. Hajlasz and S. Zimmerman , Geodesics in the Heisenberg group, Anal. Geom. Metr. Spaces, 3(2015), pp. 325–337.[13]
J. Heinonen and P. Koskela , Quasiconformal maps in metric spaces with controlled geometry,Acta Math., 181 (1998), pp. 1–61.[14]
S. Keith and X. Zhong , The Poincar´e inequality is an open ended property, Ann. of Math., 167(2008), pp. 575–599.[15]
P. Koskela and P. MacManus , Quasiconformal mappings and Sobolev spaces, Studia Math.,131 (1998), pp. 1–17.[16]
E. Le Donne and E. Hakavuori , Blowups and blowdowns of geodesics in Carnot groups,arXiv:1806.09375, (2018).[17]
N. Marola, M. Miranda Jr., and N. Shanmugalingam , Characterizations of sets of finiteperimeter using heat kernels in metric spaces, Potential Anal., 45 (2016), pp. 609–633.[18]
J. Maz´on, M. Solera, and J. Toledo , The total variation flow in metric random walk spaces,Calc. Var. Partial Differential Equations, to appear, (2019).[19]
R. Monti , Some properties of Carnot-Carath´eodory balls in the Heisenberg group, Rend. LinceiMat. Appl., 11 (2000), pp. 155–167.[20]
A. Ponce , An estimate in the spirit of Poincar´es inequality, J. Eur. Math. Soc., 6 (2004), pp. 1–15.