BESOV CAPACITY AND HAUSDORFF MEASURES IN METRIC MEASURE SPACES

This paper studies Besov p-capacities as well as their relationship to Hausdorff measures in Ahlfors regular metric spaces of dimension Q for 1 < Q < p < ∞. Lower estimates of the Besov p-capacities are obtained in terms of the Hausdorff content associated with gauge functions h satisfying the decay condition R 1 0 h(t)1/(p−1) dt t <∞.


Introduction
In this paper (X, d, µ) is a proper (that is, closed bounded subsets of X are compact) and unbounded metric space.In addition, it is Ahlfors Q-regular for some Q > 1.That is, there exists a constant C = c µ such that, for each x ∈ X and all r > 0, The expressions ||u|| Bp(X) and [u] Bp(X) from (1) and (2) are called the Besov norm and the Besov seminorm of u respectively.We have (3) [u] Bp(X) = 0 if and only if u is constant µ-a.e.

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Besov spaces were studied extensively on Ahlfors Q-regular subsets of R n by Jonsson and Wallin in [JW84].In [JW84] the authors studied mainly trace and extension results.Embeddings of homogeneous Besov spaces have recently been studied by Xiao in [Xia06].
Recently Besov spaces have been used in the study of quasiconformal mappings in metric spaces and in geometric group theory.See [BP03] and [Bou07].
Capacities associated with Besov spaces were studied by Netrusov in [Net92] and [Net96], by Adams and Xiao in [AX03], and by Adams and Hurri-Syrjänen in [AHS03].Bourdon in [Bou07] studied Besov B p -capacity in the metric setting under the assumption that X is compact.He worked with functions from A p (X), the algebra of continuous functions that are in B p (X).The algebra A p (X) does not separate in general the points of X when 1 ≤ p ≤ Q < ∞.See [Bou07], [BP03], and [Bou04].
In this paper we assume that 1 < Q < p < ∞ unless stated otherwise.Under the assumption 1 < Q < p < ∞ we develop a theory of Besov B p -capacity on X and prove that this capacity is a Choquet set function.We also obtain lower bounds for the relative Besov capacity in terms of the Hausdorff content associated with gauge functions h satisfying a certain integrability condition under the additional hypothesis that X admits a weak (1, p)-Poincaré inequality with 1 ≤ p < Q < p < ∞.Some of the ideas used here follow [KM96], [KM00], [BP03], and [Bou07].

Preliminaries
In this section we present the standard notation to be used throughout this paper.Here and throughout this paper B(x, r) = {y ∈ X : d(x, y) < r} is the open ball with center x ∈ X and radius r > 0, B(x, r) = {y ∈ X : d(x, y) ≤ r} is the closed ball with center x ∈ X and radius r > 0, while S(x, r) = {y ∈ X : d(x, y) = r} is the closed sphere with center x ∈ X and radius r > 0. For a positive number λ, λB(a, r) = B(a, λr) and λB(a, r) = B(a, λr).
Under the hypothesis of Ahlfors regularity, it is known that the metric measure space (X, d, µ) is proper if and only if it is complete.Throughout this paper (X, d, µ) is assumed to be a complete and unbounded Ahlfors Q-regular metric space for some Q > 1.
Throughout this paper, C will denote a positive constant whose value is not necessarily the same at each occurrence; it may vary even within a line.C(a, b, . . . ) is a constant that depends only on the parameters a, b, . . . .Here Ω will denote a nonempty open subset of X.For E ⊂ X, the boundary, the closure, and the complement of E with respect to X will be denoted by ∂E, E, and X \ E, respectively; diam E is the diameter of E with respect to the metric d and E ⋐ F means that E is a compact subset of F .
For two sets A, B ⊂ X, we define dist(A, B), the distance between A and B, by dist(A, B) = inf For Ω ⊂ X, C(Ω) is the set of all continuous functions u : Ω → R.Moreover, for a measurable u : Ω → R, supp u is the smallest closed set such that u vanishes on the complement of supp u.We also use the spaces Let f : Ω → R be integrable.For E ⊂ Ω measurable with 0 < µ(E) < ∞, we define We say that a locally integrable function u : X → R belongs to BMO(X), the space of functions of bounded mean oscillation, if

Besov spaces
In this section we prove some basic properties of the Besov spaces B p (X) and their closed subspaces B p (Ω) and B 0 p (Ω), where Ω ⊂ X is an open set.We also present standard lemmas needed for the proofs of our main results.
We know that in the Euclidean case B p (R n ) is a reflexive Banach space and moreover, S is dense in B p (R n ) where S = S(R n ) is the Schwartz class.See [AH96, Theorem 4.1.3]and [Pee76, Chapter 3].We would like to prove similar results about reflexivity and density when (X, d, µ) is an Ahlfors Q-regular metric space with Q > 1.It is easy to see that every Lipschitz function with compact support belongs to B p (X) whenever X is proper and 1 < Q < p < ∞.
We have the following lemmas regarding the reflexivity of B p (X) and the embedding of B p (X) into BMO(X) whenever (X, d, µ) is an Ahlfors Q-regular metric space with 1 < Q, p < ∞.
Lemma 3.1.Suppose 1 < Q, p < ∞ and that X is an Ahlfors Q-regular metric space.Then B p (X) is a reflexive space.
We endow the product space L p (X, µ) × L p (X × X, ν) with the product norm.Namely, for (u, g) Clearly this product space is reflexive because it is a product of two reflexive spaces.Since B p (X) embeds isometrically into a closed subspace of this reflexive product space, we have that B p (X) is itself a reflexive space.This finishes the proof.Lemma 3.2.Suppose 1 < Q, p < ∞ and that X is an Ahlfors Q-regular metric space.There exists a constant ( and the claim follows. Remark 3.3.We notice that in the above two lemmas the claims hold for general Q and p in (1, ∞).
From now on throughout the remainder of the paper it will be assumed that 1 For an open set Ω ⊂ X we define We notice that B p (Ω) is a closed subspace of B p (X) with respect to the Besov norm, hence it is itself a reflexive space.
To prove the second assertion, for positive integers k we define the function For a measurable function u : Ω → R, we let u + = max(u, 0) and u − = min(u, 0).
Proof: The proof is similar to the proof of [Cos07, Lemma 2.2] and omitted.
Next we show that the space B 0 p (Ω) is a lattice.Lemma 3.6.If u, v ∈ B 0 p (Ω), then min(u, v) and max(u, v) are in B 0 p (Ω).Moreover, if u ∈ B 0 p (Ω) is nonnegative, then there is a sequence of nonnegative functions ϕ j ∈ Lip 0 (Ω) converging to u in B p (Ω).
Proof: It is enough to show, due to Lemma 3.5, that u + is in B 0 p (Ω) whenever u is in Lip 0 (Ω).But this is immediate, because u + ∈ Lip 0 (Ω) whenever u ∈ Lip 0 (Ω).This finishes the proof.Lemma 3.7.Let ϕ be a Lipschitz function with compact support in X.
where C depends on Q, p, c µ , the Lipschitz constant of ϕ, and the diameter of supp ϕ.
Proof: We can assume without loss of generality that ϕ ≡ 0. Let R be the diameter of supp ϕ.We choose x 0 ∈ supp ϕ such that supp ϕ ⊂ B, where B = B(x 0 , R).Let L > 0 be a constant such that |ϕ(x) − ϕ(y)| ≤ L d(x, y) for every x, y ∈ X.Note that ||ϕ|| L ∞ (X) ≤ 2LR.We also notice that and ( 7) For every x, y ∈ X we have , where From the definition of I 11 we have, since ϕ is Lipschitz with constant L, We have (10 for every y ∈ 2B, where we recall that R is the radius of B. From (9) and (10) we get Since ϕ is supported in B, it follows from the definition of I 2 that |u(y)| p |ϕ(y)| p d(x, y) 2Q dµ(x) dµ(y). Hence and since d(x, y) ≥ d(x,x0) 2 whenever x ∈ X \ 2B and y ∈ B, we get Hence From ( 8), (11), (12), and the fact that where the constant C is as required.This finishes the proof.
Lemma 3.8.Let ϕ be a Lipschitz function with compact support in X. Suppose u k is a sequence in B p (X) converging to u in B p (X).Then u k ϕ converges to uϕ in B p (X).
Proof: From Lemma 3.7, we have that u k ϕ ∈ B p (X) for every k ≥ 1 and uϕ ∈ B p (X).Moreover, Lemma 3.7 implies for every k ≥ 1, and since u k → u in B p (X), it follows that u k ϕ → uϕ in B p (X).This finishes the proof.
By an argument similar to the one from Lemma 3.7, one can show that uϕ ∈ B p ( Ω) whenever u ∈ B p (X) and ϕ ∈ Lip 0 ( Ω) satisfies (15).Moreover, in this case ||uϕ|| Bp( e Ω) ≤ C||u|| Bp(X) for all u ∈ B p (X) and the constant C > 0 can be chosen to depend only on Q, p, c µ , dist(Ω, X \ Ω), and the diameter of Ω.
Remark 3.10.It is easy to see that uϕ ∈ B p (X) whenever u, ϕ are bounded functions in B p (X).Moreover, . We repeat to some extent the argument of Lemma 3.7 with ϕ = η and u and we note that ||η|| L ∞ (X) ≤ 1. Hence (16) where and We notice that I 1 ≤ 2 p (I 10 + I 11 ), where and We have (17) Because η is supported in 2B, it follows from the definition of I 2 that in fact As in Lemma 3.7 we get From ( 16), ( 17), ( 18), ( 19), and the fact that . This finishes the proof.
We now show that every function in B p (X) can be approximated by locally Lipschitz functions in B p (X).
Proposition 3.12.Lip loc (X) ∩ B p (X) is dense in B p (X).More precisely, if u ∈ L 1 loc (X) has finite Besov p-seminorm, then there exists a sequence u ε , ε > 0, in Lip loc (X) such that: Proof: For every ε > 0 we construct a family of balls B(x i , ε) with bounded overlap that cover X and we form a c 1 /ε-Lipschitz partition of unity associated with the cover {B(x i , ε)} as in [KL02].Here c 1 = c 1 (c µ ).More precisely, we choose a family of balls B(x i , ε), i = 1, 2, . . ., such that where c 0 is the constant from (20) and such that on X.We define the approximation by setting for every x ∈ X.We notice from (20) that for every x ∈ X at most c 0 nonzero terms appear in (21).Therefore with at most 2c 0 nonzero terms appearing in the infinite sum for every (x, y) ∈ X × X.Consequently, we obtain where c 0 is the bounded overlap constant appearing in (20).However, from Lemma 3.11 there exists a constant for every i = 1, 2, . . . .From this and (22) we obtain where we have from ( 20) and ( 23) that

An application of Lebesgue Dominated Convergence Theorem yields
for every ε > 0.
(ii) By using (20) and the fact that ϕ i forms a partition of unity we obtain, via an argument similar to the one from Lemma 3.2 where c 0 is the constant from (20).This implies immediately that ||u ε − u|| L p (X) → 0 as ε → 0. This finishes the proof.
Proof: Let u ∈ B p (X).Without loss of generality we can assume that u is bounded and locally Lipschitz.We fix x 0 ∈ X.For every integer k ≥ 2, we define ϕ k : X → R by

S ¸. Costea
We also have as k → ∞.This finishes the proof.
Proof: The proof is similar to the proof of [Cos07, Lemma 2.10].We present it for the convenience of the reader.For the proof of (i), let ψ ∈ Lip 0 (Ω) be such that ψ = 1 on the support of v.If a sequence v j ∈ Lip 0 (X) converges to v in B p (X), then from Lemma 3.8 we see that ψv j ∈ Lip 0 (Ω) converges to ψv = v in B p (X), therefore v ∈ B 0 p (Ω).As to assertion (ii), let ϕ j ∈ Lip 0 (Ω) be an approximating sequence for u ∈ B 0 p (Ω).From Lemma 3.6 we can assume that the functions ϕ j are nonnegative.We can assume without loss of generality that v = u = 0 everywhere on X \ Ω.Then min(v, ϕ j ) has as support a compact subset of Ω and hence belongs to Lemma 3.15.Suppose that Ω ⋐ X.Let u ∈ B p (Ω) be such that u = 0 on X \ Ω and lim Ω∋x→y u(x) = 0 for all y ∈ ∂Ω.Then u ∈ B 0 p (Ω).

Relative Besov capacity
In this section, we establish a general theory of the relative Besov capacity and study how this capacity is related to Hausdorff measures.
For E ⊂ Ω we define We call BA(E, Ω) the set of admissible functions for the condenser (E, Ω).The relative Besov p-capacity of the pair (E, Ω) is denoted by Since B 0 p (Ω) is closed under truncations from below by 0 and from above by 1 and since these truncations do not increase the Besov p-seminorm, we may restrict ourselves to those admissible functions u for which 0 ≤ u ≤ 1.
Remark 4.1.If K is a compact subset of the bounded and open set Ω ⊂ X, we get the same Besov B p -capacity for (K, Ω) if we restrict ourselves to a smaller set of admissible functions, namely Indeed, let u ∈ BA(K, Ω); we may clearly assume that u = 1 in a neighborhood U ⋐ Ω of K. Then we choose a cut-off Lipschitz function η, Proof: The proof is very similar to the proof of [Cos07, Theorem 3.1] and is therefore omitted.
A set function that satisfies properties (i), (iv) and (v) is called a Choquet capacity (relative to Ω).We may thus invoke an important capacitability theorem of Choquet and state the following result.See [Doo84, Appendix II].
Theorem 4.3.Suppose (X, d, µ) is a metric measure space as in Theorem 4.2.Suppose that Ω is a bounded open set in X.The set function E → cap Bp (E, Ω), E ⊂ Ω, is a Choquet capacity.In particular, all Borel subsets (in fact, all analytic) subsets E of Ω are capacitable, i.e., 4.2.Upper estimates for the relative Besov capacity.Next we derive some upper estimates for the relative Besov capacity.Similar estimates have been obtained earlier by Bourdon in [Bou07].We follow his methods.
Theorem 4.4.Let (X, d, µ) be a metric measure space as in Theorem 4.2.There exists a constant C = C(Q, p, c µ ) > 0 depending only on Q, p and c µ such that for every 0 < r < R 2 and every x 0 ∈ X. Proof: We use the function u : X → R, Then u ∈ B p (X) because it is Lipschitz with compact support.Since u is continuous on X and 0 outside B(x 0 , R), we have in fact from Lemma 3.15 that u ∈ B 0 p (B(x 0 , R)).In fact u ∈ BA(B(x 0 , r), B(x 0 , R)) since u = 1 on B(x 0 , r).Let v(x) = u(x) log R r .We will get an upper bound for [v] Bp(B(x0,R)) .Let k ≥ 3 be the smallest integer such that 2 k−1 r ≥ R. For i = 1, . . ., k we define We also define B 0 = B(x 0 , r) and B k+1 = X \ B(x 0 , 2 k r).We have Obviously we have I i,j = I j,i .We majorize I i,j by distinguishing a few cases.For j ≤ k and 0 ≤ i ≤ j − 2 we have from the definition of v that |v(x) − v(y)| ≤ j − i + 1 whenever x ∈ B i and y ∈ B j , hence

Moreover, we have
for every y ∈ B(x 0 , 2 i r), where C 2 depends only on p, Q and c µ .Hence for 0 ≤ i ≤ j ≤ k we have In particular, for j − 1 ≤ i ≤ j ≤ k, the integral I i,j is bounded by a constant that depends only on p, Q and c µ .Now we have to bound I i,j when j = k + 1.Since v is constant on B k ∪ B k+1 , we have I i,k+1 = 0 for i ∈ {k, k + 1}.For 0 ≤ i ≤ k − 1 we have But there exists C 5 > 0 such that The last sum is equal to But k + 2 − l ≤ k + 1 and there exists a > 1 such that l p 2 −lQ ≤ C 9 a −l for l ≥ 1. Hence For each (m, α) and each l < m there is a unique β such that K α m,r ⊂ K β l,r .
(iv) For every (m, α) there exists a ball B α m,r = B(x α m,r , 10 −m r) such that We call these open sets "dyadic cubes".Two distinct dyadic cubes K, K ′ in D r are adjacent if there exists an integer k such that either (i) K, K ′ are in generation k and K ∩ K ′ = ∅, or (ii) one of the cubes K, K ′ is in generation k, the other one is in generation k + 1 the one in generation k contains the other one.
Similarly, if K 0 ⊂ X is a dyadic cube in D r , we denote by D r (K 0 ) the dyadic subcubes of K 0 .
For two adjacent cubes K, K ′ ∈ D r we have where C is a constant that depends only on the Ahlfors regularity of X.
Lemma 4.5.There exists a constant C depending only on the Ahlfors regularity of X such that or µ-a.e.η, ζ ∈ X.
We also have (see [BP03, Theorem 3.4]): Lemma 4.6.There exists a constant C depending only on p and on the Ahlfors regularity of X such that for every f ∈ B p (X).
This implies (see [BP03, Lemma 3.5]): Lemma 4.7.There exists a constant C depending only on p and on the Ahlfors regularity of X such that for every f ∈ B p (X).
4.3.Hausdorff measure and relative Besov capacity.Now we examine the relationship between Hausdorff measures and the B p -capacity.
Let h be a real-valued and increasing function on [0, ∞) such that lim t→0 h(t) = h(0) = 0 and lim t→∞ h where the infimum is taken over all coverings of E by open sets G i in Ω with diameter r i not exceeding δ.The set function Λ ∞ h,Ω is called the h-Hausdorff content relative to Ω. Clearly Λ δ h,Ω is an outer measure for every δ ∈ (0, ∞] and every open set Ω ⊂ X.We write Λ δ h (E) for Λ δ h,X (E).Moreover, for every E ⊂ Ω, there exists a Borel set E such that E ⊂ E ⊂ Ω and Λ δ h,Ω (E) = Λ δ h,Ω ( E). Clearly Λ δ h,Ω (E) is a decreasing function of δ.It is easy to see that Λ δ h,Ω2 (E) ≤ Λ δ h,Ω1 (E) for every δ ∈ (0, ∞] whenever Ω 1 and Ω 2 are open sets in X such that E ⊂ Ω 1 ⊂ Ω 2 .This allows us to define the h-Hausdorff measure relative to E).The measure Λ h,Ω is Borel regular; that is, it is an additive measure on Borel sets of Ω and for each E ⊂ Ω there is a Borel set G such that E ⊂ G ⊂ Ω and Λ h,Ω (E) = Λ h,Ω (G).(See [Fed69,p. 170] and [Mat95, Chapter 4].)If h(t) = t s , we write Λ s for Λ t s ,X .It is immediate from the definition that Λ s (E) < ∞ implies Λ u (E) = 0 for all u > s.The smallest s ≥ 0 that satisfies Λ u (E) = 0 for all u > s is called the Hausdorff dimension of E.
For Ω ⊂ X open and δ > 0 the set function Λ δ h,Ω has the following property: (i) If K i is a decreasing sequence of compact sets in Ω, then Moreover, if Ω ⋐ X and h is a continuous measure function, then Λ δ h,Ω satisfies the following additional properties: (ii) If E i is an increasing sequence of arbitrary sets in Ω, then We have the following proposition: , is decreasing and there exists a constant C = C(Q, c µ ) such that for all E ⊂ X and all δ > 0 The proof is similar to the proof of [AH96, Proposition 5.1.8]and omitted.
Theorem 4.9.Suppose 1 ≤ p < Q < p < ∞.Let (X, d, µ) be a complete and unbounded Ahlfors Q-regular metric space that supports a weak (1, p)-Poincaré inequality.Suppose h : [0, ∞) → [0, ∞) is a continuous increasing measure function such that t → h(t)t −Q , 0 < t < ∞, is decreasing.Let K 0,r ∈ D r be a dyadic cube of generation 0 and let x 0 ∈ X be such that B(x 0 , r/10) ⊂ K 0,r .There exists a positive constant for every E ⊂ X, every k > 1, r > 0, and for every From the fact that there exists a Borel set E such that E ⊂ E ⊂ X and cap Bp (E ∩ K k,r , B(x 0 , r/10)) = cap Bp ( E ∩ K k,r , B(x 0 , r/10)), we can assume that E is a Borel set.All the sets E ∩ K k,r considered here lie in 7B(x 0 , 10 −k r) ⋐ B(x 0 , r/10) whenever k ≥ 2. Indeed, K k,r , which is contained in a ball of radius 10 −k 3r, was chosen such that K k,r ∩ B(x 0 , 10 −k r) = ∅.From this observation, the discussion before Proposition 4.8, and the fact that cap Bp (•, B(x 0 , r/10)) is a Choquet capacity, it follows that it is enough to consider only compact sets E in order to prove the theorem.
There is nothing to prove if either Λ ∞ h,K0,r (E ∩ K k,r ) = 0 or if So we can assume without loss of generality that α = Λ ∞ h,K0,r (E ∩ K k,r ) > 0 and that For every ζ ∈ S(x 0 , r/10) there exists an increasing sequence (K s,ζ ) s≤0 of dyadic subcubes of K 0,r such that K s,ζ is a cube of generation s for every integer s ≤ 0 and s≤0 K s,ζ = {ζ}.We denote by s 0 ζ the sequence (K s,ζ ) s≤0 .For every η ∈ K k,r there exists a decreasing sequence (K s+k,η ) s≥0 of dyadic subcubes of K k,r such that K s+k,η is of generation s + k for every s ≥ 0 and s≥0 K s+k,η = {η}.We denote by s 1 η the sequence (K s+k,η ) s≥0 .Let I = {K 0,r , . . ., K k,r } be a shortest decreasing sequence of cubes connecting K 0,r and K k,r .
For (ζ, η) ∈ S(x 0 , r/10) × K k,r we define γ ζ,η = (K s,ζ,η ) s∈Z , where For K, K ′ ∈ D r we define We notice that C(K, K ′ ) = ∅ if K, K ′ are not adjacent or if they are adjacent but of the same generation.Since X is an Ahlfors Q-regular complete metric space that satisfies a weak (1, p)-Poincaré inequality with 1 ≤ p < Q, there exists (see [Kor07, Theorem 4.2]) a constant C depending only on p and on the data of X such that for all closed spheres S(x, t) of radius t in X.We also have α = Λ ∞ h (E ∩ K k,r ) > 0. Therefore, by applying Frostman's lemma (see [Mat95,Theorem 8.8]), there exists a constant C > 0 and probability measures ν 0 on S(x 0 , r/10) and ν 1 on E ∩ K k,r such that for every ball B(x, t) of radius t in X we have and ν 1 (B(x, t)) ≤ C h(t) α .
For K, K ′ ∈ D r we define We notice that m(K, K ′ )m(K ′ , K) = 0 for every pair of cubes K, K ′ ∈ D r .Moreover, if m(K, K ′ ) = 0, then this implies that K and K ′ are adjacent but of different generations.Let f be in BW (E, B(x 0 , r/10)).Then, since f is continuous, we have that for every y ∈ X for every nested sequence K v of r-dyadic cubes containing y and converging to y.It follows that We obtain with the definition of m(K, K ′ ) and by Hölder's inequality, that where we used (26) for the last inequality.Here the constant C depends only on p and on the Ahlfors regularity of X.For a nonnegative integer s we let and similarly We notice that we can break We recall that I = {K 0,r , . . ., K k,r } is a shortest decreasing sequence of cubes in D r connecting K 0,r and K k,r .Thus, the sum in the middle is exactly k.We get upper bounds for the first and the third term in the sum.We notice that for every s ≥ 0 we have since ν 0 × ν 1 is a probability measure.On the other hand, there exists a constant C ′ depending only on p and on the Hausdorff dimension of X such that for every integer s ≥ 0 and But there exists a constant for every r > 0, every integer k > 1 and every continuous increasing measure function

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From a similar computation we get

So we get
It is easy to see that there exists a constant C depending only on p and on the Hausdorff dimension of X such that Λ ∞ h (K k,r ) for every r > 0, every integer k > 1 and every continuous increasing measure function Therefore we obtain for every integer k > 1 and for every f ∈ BW (E ∩ K k,r , B(x 0 , r/10)).This implies that there exists a constant C ′ 1 depending only on p and on the Hausdorff dimension of X such that Λ ∞ h,K0,r (E ∩ K k,r )

This finishes the proof since Λ
).As a consequence of Theorem 4.9, we obtain the following theorem.
Theorem 4.10.Suppose (X, d, µ) is a metric measure space as in Theorem 4.9.Suppose h : [0, ∞) → [0, ∞) is a continuous increasing measure function such that t → h(t)t −Q , 0 < t < ∞, is decreasing.There exists a positive constant for every E ⊂ X, every x ∈ X, and every pair of positive numbers r, R such that r < R 2 .
Proof: Fix x ∈ X and r, R such that 0 < r < R 2 .Without loss of generality we can assume that B(x, 100R) ⊂ K 0,1000R .We choose k ≥ 3 integer such that 10 2−k R ≤ r < 10 3−k R. From the construction of the dyadic cubes and the fact that X is a Q-Ahlfors regular space with Q > 1, it follows that there exists a constant C = C(Q, c µ ) independent of k such that every ball of radius 10 2−k R intersects with at most C dyadic subcubes of K 0,1000R from the kth generation.We leave the rest of the details to the reader.
It follows easily that if X is a complete and unbounded Ahlfors Q-regular metric space as in Theorem 4.9, then there exists a constant whenever E ⊂ X, R > 0, and a ∈ X.
As a corollary we have the following.
Corollary 4.11.Suppose (X, d, µ) is a metric measure space as in Theorem 4.9.There exists a positive constant C 2 = C 2 (Q, p, p, c µ ) such that for every x ∈ X and every pair of positive numbers r, R such that r < R 2 .
Proof: We apply Theorem 4.10 for h(t) = t Q−e p .We notice (see [Kor07,Theorem 4.2]) that there exists a constant for every x ∈ X and every r > 0. The rest is routine.
Theorem 4.12.Suppose (X, d, µ) is a metric measure space as in Theorem 4.9.There exists for every x ∈ X and every pair of positive numbers r, R such that r < R 2 .A set E ⊂ X is said to be of Besov B p -capacity zero if cap Bp (E ∩ Ω, Ω) = 0 for all open and bounded Ω ⊂ X.In this case we write cap Bp (E) = 0.The following lemma is obvious.Lemma 4.14.Suppose that E is bounded and that there is a bounded neighborhood Ω of E with cap Bp (E, Ω) = 0. Then cap Bp (E) = 0.
Proof: The proof is similar to the proof of [Cos07, Lemma 3.13] and omitted.
Corollary 4.15.Suppose (X, d, µ) is a metric measure space as in Theorem 4.9.Let E ⊂ X be such that cap Bp (E) = 0. Then Λ h (E) = 0 for every measure function h In particular, the Hausdorff dimension of E is zero and X \ E is connected.
Proof: To prove the first claim, it is enough to assume, without loss of generality, that h : [0, ∞) → [0, ∞) is a continuous measure function such that t → h(t)t −Q , 0 < t < ∞, is decreasing.(See Proposition 4.8.)If cap Bp (E) = 0, then there exists a Borel set E such that E ⊂ E and cap Bp ( E) = 0, hence we can assume without loss of generality that E is itself Borel.Since Λ ∞ h is a countably subadditive set function and Λ h (E) = 0 if and only if Λ ∞ h (E) = 0 whenever h is a continuous measure function, it is enough to assume that E is bounded.Moreover, the discussion before Proposition 4.8 shows that it is enough to assume that E is in fact compact.For E compact the first claim follows obviously from Theorem 4.10.The second claim is a consequence of the first claim because for every s ∈ (0, Q), the function h s : [0, ∞) → [0, ∞) defined by h s (t) = t s has the property (33).The third claim is a consequence of the Poincaré inequality.
We also get upper bounds of the relative Besov p-capacity in terms of a certain Hausdorff measure.
Proposition 4.16.Let h : [0, ∞) → [0, ∞) be an increasing homeomorphism such that h(t) = (log 1 t ) 1−p for all t ∈ (0, 1 2 ).Suppose (X, d, µ) is a proper and unbounded Ahlfors Q-regular metric space.Let E be a compact subset of X.There exists a constant C depending only on p and on the Ahlfors regularity of X such that cap Bp (E, Ω) ≤ CΛ h (E) for every bounded and open set Ω containing E.
Proof: The proof is similar to the proof of [Cos07, Proposition 3.17].We present it for the convenience of the reader.We can assume without loss of generality that Λ h (E) < ∞.Let Ω be a bounded open set containing E. We denote by δ the distance from E to the complement of Ω.Without loss of generality we can assume that 0 < δ < 1.We fix 0 < ε < 1 such that 0 < ε < δ 2 4 .Then r < ε implies log δ 2r ≥ 1 2 log 1 r .We cover E by finitely many open balls B(x i , r i ) such that r i < ε 2 .Since we may assume that the balls B In the last step we also used formula (32) for the Besov B p -capacity of spherical condensers together with our choice of ε.Taking the infimum over all such coverings and letting ε → 0, we conclude cap Bp (E, Ω) ≤ CΛ h (E).This finishes the proof of the proposition.
Proposition 4.16 gives another sufficient condition to obtain sets of Besov p-capacity zero.
Proof: The proof is similar to the proof of [Cos07,Theorem 3.16].Since Λ h is a Borel regular measure, we may assume that E is a Borel set and furthermore, in light of the Choquet capacitability theorem, we may assume that E is compact.We let M = CΛ h (E), where C is the constant from Proposition 4.16.Since Λ h (E) < ∞, we have that µ(E) = 0, while Proposition 4.16 implies that cap Bp (E, Ω) ≤ M for every bounded and open set Ω containing E. Let Ω ⊂ X be a bounded open set containing E. From Lemma 4.14 it is enough to show that cap Bp (E, Ω) = 0. We choose a descending sequence of bounded open sets Bp(Ωi) < M + 1.Then ϕ i is a bounded sequence in B p (Ω).Because ϕ i converges pointwise to a function ψ which is 0 in X \ E and 1 on E, we have from Mazur's lemma and the reflexivity of B 0 p (Ω) that ψ ∈ B 0 p (Ω).That is, there exists a subsequence denoted again by ϕ i such that ϕ i → ψ weakly in B 0 p (Ω) and a sequence ϕ i of convex combinations of ϕ j , Without loss of generality we can assume that ϕ i → ψ pointwise in X as i → ∞.The convexity of the Besov seminorm and the choice of the sequence ϕ i imply, together with the closedness of BW (E, Ω i ) under finite convex combinations, that ϕ i ∈ BW (E, Ω i ) for every integer i ≥ 1.Since µ(E) = 0, ψ = 0 in X \ E, and ϕ i → ψ in B 0 p (Ω), it follows that ||ψ|| Bp(Ω) = 0.This implies Bp(Ω) = 0.This finishes the proof.

Besov capacity and quasicontinuous functions
In this section we study a global Besov capacity and quasicontinuous functions in Besov spaces.
Theorem 5.10.Let ϕ j ∈ C(X)∩B p (X) be a Cauchy sequence in B p (X).Then there is a subsequence ϕ k which converges B p -quasiuniformly in X to a function u ∈ B p (X).In particular, u is B p -quasicontinuous and ϕ k → u B p -quasieverywhere in X.
Proof: The proof is similar to the proof of [HKM93, Theorem 4.3] and omitted.
Theorem 5.10 implies the following corollary.
Corollary 5.11.Suppose that u ∈ B p (X).Then there exists a B p -quasicontinuous Borel function v ∈ B p (X) such that u = v µ-a.e.
Proof: Since u ∈ B p (X), from Theorem 3.13 there exists a sequence of functions ϕ j in Lip 0 (X) converging to u in B p (X). Passing to subsequences if necessary, we can assume that ϕ j → u pointwise µ-a.e. in X and that 2 jp ||ϕ j+1 − ϕ j || p L p (X) + [ϕ j+1 − ϕ j ] p Bp(X) < 2 −j for every j = 1, 2, . . . .Defining E j = {x ∈ X : |ϕ j+1 − ϕ j | > 2 −j } and letting E = ∩ ∞ k=1 ∪ j=k E j , the proof of Theorem 5.10 yields the existence of a function v ∈ B p (X), such that ϕ j → v in B p (X) and pointwise in X \ E. Since E is a Borel set of Besov B p -capacity zero and the functions ϕ j are continuous, this finishes the proof.
Theorem 5.12.Let u ∈ B p (X).Then u ∈ B 0 p (Ω) if and only if there exists a B p -quasicontinuous function v in X such that u = v µ-a.e. in Ω and v = 0 q.e. in X \ Ω.
Proof: Fix u ∈ B 0 p (Ω) and let ϕ j ∈ Lip 0 (Ω) be a sequence converging to u in B p (Ω).By Theorem 5.10 there is a subsequence of ϕ j which converges B p -quasieverywhere in X to a B p -quasicontinuous function v in X such that u = v µ-a.e. in Ω and v = 0 q.e. in X \ Ω. Hence v is the desired function.
To prove the converse, we assume first that Ω is bounded.Because the truncations of v converge to v in B p (Ω), we can assume that v is bounded.Without loss of generality, since v is B p -quasicontinuous and v = 0 q.e.outside Ω we can assume that in fact v = 0 everywhere in X \ Ω. Choose open sets G j such that v is continuous on X \ G j and Cap Bp (G j ) → 0. By passing to a subsequence, we may pick a sequence ϕ j in B p (X) such that 0 ≤ ϕ j ≤ 1, ϕ j = 1 everywhere in G j , ϕ j → 0 µ-a.e. in X, and ||ϕ j || p L p (X) + [ϕ j ] p Bp(X) → 0.
Then from Remark 3.10 we have that w j = (1 − ϕ j )v is a bounded sequence in B p (Ω).Moreover, for every j ≥ 1, we have lim x→y,x∈Ω w j (x) = 0 for all y ∈ ∂Ω.Thus, from Lemma 3.15, we have that w j is a sequence in B 0 p (Ω). Clearly w j → v in L p (X) and pointwise µ-a.e. in X.This, together with the boundedness of the sequence w j in B 0 p (Ω), implies via Mazur's lemma that v ∈ B 0 p (Ω).The proof is complete in case Ω is bounded.
We denote by the set of all functions u ∈ B p (X) such that there exists a sequence ϕ j ∈ C(X) ∩ B p (X) converging to u both in B p (X) and B p -quasiuniformly.It follows immediately from Theorem 5.10 that the functions in Q Bp are B p -quasicontinuous and for each v ∈ B p (X) there is u ∈ Q Bp such that u = v µ-a.e.We soon show that, conversely, each B p -quasicontinuous function v of B p (X) belongs to Q Bp .Proof: The proof is similar to the proof of [HKM93, Lemma 4.7] and omitted.
This result has the following corollary.
Corollary 5.14.Suppose that Ω is open and bounded and let E ⋐ Ω.Let u ∈ Q Bp .Suppose that u ≥ 1 quasieverywhere on E and that u has compact support in Ω.Then cap Bp (E, Ω) ≤ [u] p Bp(Ω) .We know that Cap Bp is an outer capacity.It satisfies the following compatibility condition (see [Kil98]): Thus, u is a B p -quasicontinuous Borel function and u = u µ-a.e.This proves the existence of a B p -quasicontinuous Borel representative of u.The claim follows.

.
The claim follows with C = C 10 .For a fixed r > 0 we construct the dyadic partition of X as in [Chr90, Theorem 11].That is, a family of open sets D r = {K α m,r : m
Theorem 4.2.Suppose (X, d, µ) is a proper and unbounded Ahlfors Q-regular metric space with 1 < Q < p < ∞.Let Ω ⊂ X be a bounded open set.The set function E → cap Bp (E, Ω), E ⊂ Ω, enjoys the following properties: .1.Basic properties of the relative Besov capacity.A capacity is a monotone, subadditive set function.The following theorem expresses, among other things, that this is true for the relative Besov p-capacity.
Lemma 4.13.A countable union of sets of Besov B p -capacity zero has Besov B p -capacity zero.The next lemma shows that, if E is bounded, one needs to test only a single bounded open set Ω containing E in showing that E has zero Besov B p -capacity.