Maximal function estimates and self-improvement results for Poincar\'e inequalities

Our main result is an estimate for a sharp maximal function, which implies a Keith-Zhong type self-improvement property of Poincar\'e inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces.


Introduction
Relatively standard assumptions in analysis on metric measure spaces are a doubling condition on the measure and a Poincaré type inequality for a certain class of functions. Roughly speaking a Poincaré inequality transfers infinitesimal information encoded in the derivative to larger scales. It also relates the notion of a derivative to the given measure and, together with the doubling condition, implies Sobolev inequalities. We consider the so-called D-structures introduced in [5], which give a very general notion of a derivative with natural differentiation properties in metric measure spaces. This gives an axiomatic point of view to the theory of Sobolev spaces on metric measure spaces, which includes the standard maximal and upper gradient approaches studied, for example, in [6,14]. Standard references to analysis on metric measure spaces are [1,9,10].
Keith and Zhong proved in [12] that Poincaré inequalities are self-improving under certain assumptions. More precisely, their result improves a (1, p)-Poincaré inequality with p > 1 to a (1, p − ε)-Poincaré inequality for some ε > 0. This open ended property is of fundamental importance not only because of its theoretical interest but also because of its applications, for example, to regularity theory in the calculus of variations, we refer to [12] and references therein. In this work we establish a corresponding self-improvement property for D-structures, see Theorem 5.8 below. Our goal is to give an abstract and transparent argument with a special emphasis on the role of the underlying space and relevant maximal function inequalities. Indeed, instead of a good lambda inequality [12, Proposition 3.1.1], our main result Theorem 4.3 gives a new estimate for the sharp maximal function associated with a given D-structure. This result may be of independent interest and several questions related to weighted norm inequalities for future research arise.
A distinctive feature of our approach is that, in addition to the standard Lipschitz scale, we also consider Hölder continuous functions. Moreover, the role of the underlying space is visible only by way of the D-structure and certain geodesic arguments. On technical level our argument differs from that of [12] in the sense that Whitney type extension theorems for Lipschitz functions are completely avoided and the stopping time argument is tailored for D-stuctures. We would also like to point out that there is only one single place in the proof of Theorem 4.3 where the assumed Poincaré inequality is needed. Another approach to the Keith-Zhong theorem has been recently given in [2].
As an application of our main result we study universality results for Sobolev spaces related to D-structures. More precisely, Theorem 6.2 gives a D-structure independent representation for the Sobolev norm. We also show that any abstract Sobolev space, rising from a suitable Dstructure, is isomorphic to one particular Sobolev space. This extends and complements results in [14,15].

Tracking constants
Our results are based on quantitative estimates and absorption arguments, where it is often crucial to track the dependencies of constants quantitatively. For this purpose, we will use the following notational convention: C( * , · · · , * ) denotes a positive constant which quantitatively depends on the quantities indicated by the * 's but whose actual value can change from one occurrence to another, even within a single line.

Metric spaces
Here, and throughout the paper, we assume that X = (X, d, µ) is a metric measure space equipped with a metric d and a positive complete Borel measure µ such that 0 < µ(B) < ∞ for all balls B ⊂ X, each of which is always an open set of the form B = B(x, r) = {y ∈ X : d(y, x) < r} with x ∈ X and r > 0. As in [1, p. 2], we extend µ as a Borel regular (outer) measure on X. We remark that the space X is separable under these assumptions, see [1,Proposition 1.6]. We also assume that #X ≥ 2 and that the measure µ is doubling, that is, there is a constant c µ > 1, called the doubling constant of µ, such that for all balls B = B(x, r) in X. Here we use for 0 < t < ∞ the notation tB = B(x, tr). In particular, for all balls where s = log 2 c µ > 0. We refer to [9, p. 31].

Geodesic spaces
Let X be a metric space satisfying the conditions stated in §2.2. By a curve we mean a nonconstant, rectifiable, continuous mapping from a compact interval of R to X; we tacitly assume that all curves are parametrized by their arc-length. We say that X is a geodesic space, if every pair of points in X can be joined by a curve whose length is equal to the distance between the two points. In particular, it easily follows that for all balls B = B(x, r) in a geodesic space X.
Lemma 2.4. Suppose that X is a geodesic space and A ⊂ X is a measurable set. Then the function then follows from the doubling condition (2.1) and the fact that B ′ ⊂ B(y, 2r ′ ).

Hölder and Lipschitz functions
Let A ⊂ X. We say that u : A → R is a β-Hölder function, with an exponent 0 < β ≤ 1 and a constant 0 for all x, y ∈ A .
If u : A → R is a β-Hölder function, with a constant κ, then the classical McShane extension defines a β-Hölder function v : X → R, with the constant κ, which satisfies v| A = u; we refer to [9, pp. 43-44]. The set of all β-Hölder functions u : A → R is denoted by Lip β (A). The 1-Hölder functions are also called Lipschitz functions. We denote Lip(A) = Lip 1 (A).

Definition and basic properties of D-structures
We adapt the terminology from [5] concerning the so-called D-structures. This structural framework captures the properties that we will need for Keith-Zhong type self-improvement of Poincaré inequalities, treated in §4- §6. In the following definition, and throughout the paper, we use the following familiar notation: is the integral average of u ∈ L 1 (A) over a measurable set A ⊂ X with 0 < µ(A) < ∞. Moreover if E ⊂ X, then 1 E denotes the characteristic function of E; that is, Definition 3.1. Let X be a metric measure space (recall §2.2). Fix 1 ≤ p < ∞ and 0 < β ≤ 1. Suppose that for each u ∈ Lip β (X), we are given a family D(u) = ∅ of measurable functions X → [0, ∞] as follows. First, we assume the following Poincaré inequality condition: (D1) There are constants K > 0 and τ ≥ 1 such that the (1, p)-Poincaré inequality holds whenever B is a ball in X and whenever u ∈ Lip β (X) and g ∈ D(u). Second, for all β-Hölder functions u, v : X → R, we assume the following conditions (D2)-(D4): (D2) |a|g ∈ D(au) if a ∈ R and g ∈ D(u); If v : X → R is β-Hölder with a constant κ ≥ 0 and v| X\E = u| X\E for a Borel set E ⊂ X, then κ1 E + g u 1 X\E ∈ D(v) whenever g u ∈ D(u). Then we say that the family {D(u) : u ∈ Lip β (X)} is a D-structure in X, with exponents 1 ≤ p < ∞ and 0 < β ≤ 1, and with constants K > 0 and τ ≥ 1.
Later in §4 we will need a stronger form of the condition (D1). This stronger form (D1'), corresponding to a (p, p)-Poincaré inequality, is explicitly stated in the following theorem. Theorem 3.3. Suppose that {D(u) : u ∈ Lip β (X)} is a D-structure in a geodesic space X, with exponents 1 ≤ p < ∞ and 0 < β ≤ 1, and with constants K > 0 and τ ≥ 1. Then the following condition is valid: (D1') There exists K p,p = C(c µ , β, p, q, τ )K > 0 such that the (p, p)-Poincaré inequality holds whenever B is a ball in X and whenever u ∈ Lip β (X) and g ∈ D(u).
Theorem 3.3 is an immediate consequence of a stronger result, namely the D-structure independent Theorem 3.6. Moreover, the latter result gives (q, p)-Poincaré inequalities for some q > p. By formulating Theorem 3.6 separately, we wish to emphasize the contrast that Dstructures are not needed in this 'simpler' aspect of self-improvement.
Lemma 3.4. Suppose that X is a geodesic space and that τ ≥ 1. Then there are constants M = C(τ ) ≥ 1 and a = C(τ ) > 1 as follows. Every ball B ⊂ X contains a ball B 0 ⊂ B such that, for each x ∈ B, there is a sequence of balls {B i : i = 1, 2, . . .} in X satisfying the following conditions: We also need the following lemma, which is essentially [9,Lemma 4.22]. See also [3, p. 485].
Lemma 3.5. Let B ⊂ X be a ball in a metric space and let u : B → R be a measurable function. Fix 1 ≤ q < t < ∞ and C 0 > 0 such that The following self-improvement result follows from a straightforward adaptation of the main result in [8] that corresponds to the case β = 1. We refer to [4] for versions of this result taking place in general metric spaces and with any β > 0. For convenience, we recall the proof. Theorem 3.6. Suppose that X is a geodesic space. Fix exponents 1 ≤ p < ∞ and 0 < β ≤ 1. Suppose that u ∈ Lip β (X) and that g : X → [0, ∞] is a measurable function. Assume further that there are constants K > 0 and τ ≥ 1 such that inequality where c µ is the doubling constant of µ. Fix 1 ≤ q < Qp/(Q − βp). Then there is a constant C = C(c µ , Q, β, p, q, τ ) > 0 such that inequality Proof. Fix u, g and a ball B = B(x 0 , r) ⊂ X with r > 0. Without loss of generality, we may assume that r ≤ 2 diam(B). Let B 0 ⊂ B be the fixed ball as in Lemma 3.4 for the given B ⊂ X and τ ≥ 1. By subtracting a constant from u, if necessary, we can assume that u B 0 = 0.
Let λ > 0 and let x ∈ U λ = {y ∈ B : |u(y)| > λ}. Fix {B i = B(x i , r i ) : i = 1, 2, . . .} and {R i : i = 0, 1, . . .} that are associated with the point x and the ball B as in Lemma 3.4. In particular, the properties (a)-(d) of the chain are valid. By the properties (b) and (c), we have u B i → u(x) as i → ∞, and so Hence for any 0 < ε < 1, that is to be chosen later, we obtain that By comparing the sums on the left and right, we obtain an index i x ∈ {0, 1, . . .} such that By the previous estimates and property (a) of the chain, (3.7) The assumptions on Q, inequality (2.2), and properties (a) and (c) together imply that By first raising this to power βp(ε − 1)/Q < 0 and then substituting the result to (3.7), Using the 5r-covering lemma [1, Lemma 1.7], we obtain a countable and disjoint subfamily x k holds true. Let us also observe that 0 < 1 + βp(ε − 1)/Q < 1. Hence, by the above covering property and (3.8), (3.9) Recall that βp < Q and 1 ≤ q < Qp/(Q − βp). These facts allows us to choose the number 0 < ε < 1, depending on Q, p and β only, such that max{q, p} < t = p/(1 + βp(ε − 1)/Q). Thus, by raising inequality (3.9) to the power t/p and applying Lemma 3.5, we obtain Since B ⊂ X is an arbitrary ball, we conclude the proof by raising both sides to power 1/t and recalling that r ≤ 2 diam(B).

The main result
Here we formulate and prove our main result, Theorem 4.3. This theorem can be viewed as a boundedness result for a certain maximal function which, in turn, is naturally associated with a given D-structure. More specifically, let 1 < p < ∞ and 0 < β ≤ 1. If B = ∅ is a given family of balls in X, then we define a fractional sharp maximal function We are primarily interested in the localized maximal function M ♯,p β,B 0 u that is associated with the ball family B 0 = {B ⊂ X : B is a ball such that 2B ⊂ B 0 } ; (4.2) here and in the statement of Theorem 4.3, the set B 0 ⊂ X of localization is a fixed ball, and the case X = B 0 is allowed but then X is of course necessarily bounded. Theorem 4.3. Suppose we are given a D-structure in a geodesic space X, with exponents 1 < p < ∞ and 0 < β ≤ 1. Let K p,p > 0 be the constant for the (p, p)-Poincaré inequality as in condition (D1') of Theorem 3.3. Let k ∈ N, 0 ≤ ε < p − 1, and α = βp 2 /(2(s + βp)) > 0 with s = log 2 c µ . Suppose that B 0 ⊂ X is a fixed ball. Then inequality holds for each u ∈ Lip β (X) and every g ∈ D(u). Here the constant C 1 > 0 depends only on the parameters β, p, c µ ; and C(k, ε) = (4 kε − 1)/ε if ε > 0 and C(k, 0) = k.
Let us observe that the first term on the right-hand side of (4.4) is finite, since u is assumed to be a β-Hölder function. The following corollary is obtained when this term is absorbed to the left-hand side after choosing the numbers k and 0 ≤ ε < ε 0 appropriately; for instance, we can choose ε 0 = 1/k for a large enough k.
Corollary 4.5. Suppose that we are given a D-structure in a geodesic space X, with exponents 1 < p < ∞ and 0 < β ≤ 1. Then there exists some 0 < ε 0 < p − 1 with the property that for every 0 ≤ ε < ε 0 there is a constant C > 0 such that inequality holds whenever B 0 is a ball in X and whenever u ∈ Lip β (X) and g ∈ D(u).
Question 4.7. Corollary 4.5 suggests the following problem related to weighted inequalities. Fix 1 < p < ∞ and 0 < β ≤ 1. Let us denote by B the family of all balls in X. Then, for some interesting D-structure, is it possible to characterize those weights w in X for which inequality holds for each u ∈ Lip β (X) and for every g ∈ D(u)? To our knowledge, this is an open problem even when X = R n equipped with the Lebesgue measure.
Remark 4.8. It is instructive to reflect Question 4.7 and Corollary 4.5 by considering the following simple analogy with X = R n equipped with the Lebesgue measure. If 1 < p < ∞, then the Muckenhoupt A p class consists precisely of weights w for which the maximal operator . Whereas Question 4.7 asks for a counterpart of this classical result in the present setting, Corollary 4.5, in turn, corresponds to a rather curious special case. Namely, let 0 ≤ δ < 1 and let u be a measurable function with 0 Moreover, the A p constant of this weight is independent of u. By the boundedness of the maximal function in L p (w dx), and the fact that w(x) ≤ |u(x)| −ε almost everywhere, we find that In some cases, see §5 in particular, we can further adapt this computation to the present setting.
The proof of Theorem 4.3 is completed in §4.6. For the proof, we need preparations that are treated in §4.2 - §4.5. At this stage, we already fix X, the D-structure, K p,p , B 0 X, B 0 , p, β, ε, k and u as in the statement of Theorem 4.3. We refer to these objects throughout §4 without further notice. Notice, however, that the function g is not yet fixed.
Let us emphasize that the ball B 0 in the proof below is further assumed to be a strict subset of X. That is, we will only focus on the case B 0 = X. We remark that if B 0 = X, then X is bounded and the following Whitney cover W 0 is replaced with the singleton {Q = B 0 }. The other modifications in this easier special case are straightforward and we omit the details.

Whitney ball covering
We need a Whitney ball covering W 0 = W(B 0 ) of the ball B 0 X. This countable family with good covering properties is comprised of the so-called Whitney balls that are of the form The 4-dilated Whitney ball is denoted by Q * = 4Q = B(x Q , 4r Q ) whenever Q ∈ W 0 . Even though the Whitney balls need not be pairwise disjoint, they nevertheless have the following standard covering properties with bounded overlap; cf. [1, pp. 77-78].
The facts (W3)-(W6) below for any Whitney ball Q = B(x Q , r Q ) ∈ W 0 are straightforward to verify by using inequality (2.3) and the assumption B 0 X; we omit the simple proofs. Below we refer to the family B 0 of balls that is defined in (4.2) by using the fixed ball B 0 .
Observe that there is some overlap between the properties (W4)-(W6). The slightly different formulations will conveniently guide the reader in the sequel.

Fractional sharp maximal functions
We abbreviate M ♯ u = M ♯,p β,B 0 u and denote The sets U λ are open in X. If E ⊂ X is a Borel set and λ > 0, we write U λ E = U λ ∩ E. We also need a certain smaller maximal function that is localized to Whitney balls. More specifically, for each Q ∈ W 0 , we first consider the ball family 1 By using these individual maximal functions, we then define a Whitney-ball localized sharp maximal function 2 If λ > 0 and Q ∈ W 0 , we write We need the following norm estimate between the different maximal functions. Its purpose, roughly speaking, is to create space for the forthcoming stopping balls in §4.4 to expand, without losing their control in terms of M ♯ u. On the other hand, controlling this expansion is the only purpose for introducing the different maximal functions aside from M ♯ u.
By using also the corresponding identity for the maximal function M ♯ loc u, we see that it suffices to prove that inequality holds for some C 1 = C(c µ , p, β) ≥ 1. Indeed, then one can choose C = C 1+p 1 . We will now show how inequality (4.11) follows from an adaptation of [10, Lemma 12.3.1]. However, the simple but tedious modification of the last rather short lemma is left to the interested reader.
Fix x ∈ B 0 and let us consider any ball B = B(x B , r B ) which satisfies the two conditions x ∈ B and 256B = B(x B , 256r B ) ⊂ B 0 . By the covering condition (W1) there is a Whitney ball Q = B(x Q , r Q ) ∈ W 0 such that x ∈ Q. We claim that B ⊂ Q * . In order to show this, we fix y ∈ B ⊂ B(x, 2r B ). Since B(x, 255r B ) ⊂ B 0 , we find that It follows that y ∈ 4Q = Q * . We have shown that B ⊂ Q * , and therefore x ∈ B ∈ B Q . Thus, With the aid of this estimate, the distributional inequality (4.11) follows from an adaptation of [ Proof. Let us remark that the property (W6) is used below without further notice. Fix λ > 0, Q ∈ W 0 and x, y ∈ Q * \ U λ . Write d = d(x, y). Since Q * ⊂ B 0 , it suffices to prove that (4.13) We first consider a point z ∈ Q * and a radius 0 < r ≤ 2 diam(Q * ). Write B i = B(z, 2 −i r) ∈ B 0 for each i ∈ {0, 1, . . .}. Then, with the standard 'telescoping' argument, see for instance the proof of [7, Lemma 3.6], we obtain As a consequence, since y ∈ Q * and 0 < d = d(x, y) ≤ diam(Q * ),

It follows that
which is the desired inequality (4.13).

Stopping construction
The following stopping construction is needed for each Whitney ball separately. Fix a Whitney ball Q ∈ W 0 . The number serves as a certain treshold value. Fix a level λ > λ Q /2. We will construct a stopping family S λ (Q) of balls whose 5-dilations, in particular, cover the set Q λ ; recall the definition from (4.9). As a first step towards the stopping balls, let B ∈ B Q be such that B ∩ Q = ∅. The parent ball of B is then defined to be π(B) = 2B if 2B ⊂ Q * and π(B) = Q * otherwise. Observe that B ⊂ π(B) ∈ B Q and π(B) ∩ Q = ∅ so that the grandparent π(π(B)) is well defined, and so on and so forth. Moreover, by inequalities (2.1) and (2.3), and property (W3) if needed, we have µ(π(B)) ≤ c 5 µ µ(B) and diam(π(B)) ≤ 16 diam(B). Now we come to the actual stopping argument. Let us fix a point x ∈ Q λ ⊂ Q. If λ Q /2 < λ < λ Q , then we choose B x = Q * ∈ B Q . If λ ≥ λ Q , then by using the condition x ∈ Q λ we first choose a starting ball B, with x ∈ B ∈ B Q , such that We continue by looking at the balls B ⊂ π(B) ⊂ π(π(B)) ⊂ · · · and we stop at the first ball among them, denoted by B x ∈ B Q , that satisfies the following stopping conditions: The inequality λ ≥ λ Q in combination with assumption B 0 X ensures that there always is such a stopping ball. In both cases above, the chosen ball B λ x = B x ∈ B Q contains the point x and satisfies inequalities (4.14) Now, by using the 5r-covering lemma, we obtain a countable disjoint family of stopping balls such that Q λ ⊂ ∪ B∈S λ (Q) 5B. Let us remark that, by the condition (W4) and stopping inequality (4.14), we have B ⊂ U λ Q * = U λ ∩ Q * if B ∈ S λ (Q) and λ > λ Q /2.

Auxiliary local results
We prove two technical results: Lemma 4.15 and Lemma 4.23. Even though the following lemma is a counterpart of [12, Lemma 3.1.2], the adaptation to our setting is non-trivial. Recall that k ∈ N is a fixed number and α = βp 2 /(2(s + βp)) > 0 with s = log 2 c µ > 0.

By Lemma 2.4 and the fact that B
is open, we find that h : (0, ∞) → R is continuous. Since h(r) = 1 for small values of r > 0, and h(r) < 1/2 for r > diam(B), we find that h(r x ) = 1/2 for some 0 < r Then (4.17) and (4.18) hold for every ball B ′ ∈ G λ ; indeed, by denoting we have the following comparison identities: where all the measures are strictly positive. These identities are important and they are used several times throughout the remainder of this proof. We multiply the left-hand side of (4.16) by diam(B) βp and then estimate as follows: (4.20) By (2.1) and Lemma 2.5, we find that µ( Hence, by the comparison identities (4.19), This concludes our analysis of the 'easy term' in (4.20). In order to treat the remaining term therein, we do need some preparations. Let us fix a ball B ′ ∈ G λ that satisfies In order to prove this inequality, we fix a number m ∈ R such that Let us first consider the case m < k/2. Then m − k < −k/2, and since always α < p/2, the desired inequality (4.22) is obtained in this case as follows: Next we consider the case k/2 ≤ m. By comparison identities (4.19) and Lemma 2.5, where the last step follows from condition (W5) and the fact that 5B ′ ⊃ B ′ O = ∅. It follows that 2 mp ≤ 2 p+1 c 6 µ 2 kp . On the other hand, we have where the last step follows from the fact that B ∈ S λ (Q) in combination with inequality (4.14).
In particular, if s = log 2 c µ then by inequality (2.2) and Lemma 2.5, we obtain that Combining the above estimates, we see that That is, inequality (4.22) holds also in the present case k/2 ≤ m. By using Lemma 2.5 and inequalities (4.19) and (4.22), we can now estimate the second term in (4.20) as follows: . Inequality (4.16) follows by collecting the above estimates.
The following lemma is essential for the proof of Theorem 4.3, and it is the only place in the proof where the (p, p)-Poincaré inequality is needed-and, moreover, this inequality is applied only a single time.
Lemma 4.23. Fix a Whitney ball Q ∈ W 0 . Then inequality holds for each λ > λ Q /2 and every g ∈ D(u).
Proof. Fix λ > λ Q /2 and g ∈ D(u). By the doubling condition (2.1), Recall also that B ⊂ U λ Q * = U λ ∩ Q * if B ∈ S λ (Q). Therefore, and using the fact that S λ (Q) is a disjoint family, it suffices to prove that inequality holds for every B ∈ S λ (Q). To this end, let us fix a ball B ∈ S λ (Q).
which suffices for the required local estimate (4.25). Let us then consider the more difficult case µ(U 2 k λ B ) < µ(B)/2. In this case, by the stopping inequality (4.14), By Lemma 4.15 it suffices to estimate the integral over the set B \ U 2 k λ = B \ U 2 k λ B ; observe that the measure of this set is strictly positive. We remark that the Poincaré inequality condition (D1') will be used to estimate this integral.
Fix a number i ∈ N. Recall that B ⊂ Q * . Hence, it follows from Lemma 4.12 that the restriction u| B\U 2 i λ : B \ U 2 i λ → R is a β-Hölder function with a constant κ i = C(β, c µ )2 i λ. We can now use the McShane extension (2.6) and extend u| B\U 2 i λ to a function u 2 i λ : X → R that is β-Hölder with the constant κ i and satisfies the restriction identity u 2 i λ | B\U 2 i λ = u| B\U 2 i λ .
The crucial idea that was also used by Keith-Zhong in [12] is to consider the function By conditions (D2)-(D4) of the fixed D-structure, we obtain that . By using these inclusions it is straightforward to show that the following pointwise estimates are valid in X, Observe that h coincides with u on B \ U 2 k λ and recall that g h ∈ D(h). Hence the Poincaré inequality from condition (D1') in Theorem 3.3 implies that By (W2) we have Q∈W 0 1 Q * ≤ C(c µ )1 B 0 . Hence, we can now continue to estimate as follows. First, Second, Third, by Fubini's theorem, Combining the estimates above, we arrive at the desired conclusion.

Keith-Zhong theorems
We consider the self-improvement properties of Poincaré inequalities involving p-weak upper gradients; in particular, we recover the so-called Keith-Zhong Theorem [12]; see Theorem 5.7. Moreover, we obtain a partial version of this result for D-structures in Theorem 5.8.
We refer to [1,9,10] for further information on p-weak upper gradients.
Fix an exponent 1 < p < ∞. For each function u ∈ Lip(X), we let D 1,p N (u) be the family of all p-weak upper gradients g ∈ L p loc (X) of u; by g ∈ L p loc (X) we mean that for each x ∈ X there exists r x > 0 such that g ∈ L p (B(x, r x )). The properties (D2) and (D3) in Definition 3.1 are rather well known, see for instance [1, Corollary 1.39]. The property (D4) with β = 1 is a consequence of a so-called 'Glueing lemma', we refer to [1,Lemma 2.19,Remark 2.28]. However, the (1, p)-Poincaré inequality condition (D1), with β = 1, is not always valid, and therefore we need to assume this in some of the forthcoming results.
Beyond these properties (D1)-(D4), we also need some other observations. The above family D 1,p N (u) has the following minimality property: if u ∈ Lip(X), then there exists a p-weak upper gradient g u ∈ D 1,p N (u) such that g u ≤ g almost everywhere if g ∈ D 1,p N (u); see [1,Theorem 2.25]. Moreover, this minimal p-weak upper gradient g u is unique up to sets of measure zero in X. The following result is an adaptation of [14,Lemma 4.7]; see also [11]. The proof below relies on a localization property of the minimal p-weak upper gradient to open sets (e.g. to balls B 0 in X). At this stage, the reader is encouraged to recall Definition (4.1).
Suppose that u : X → R is a Lipschitz function and let g u ∈ L p loc (X) be its minimal p-weak upper gradient. Then inequality holds for almost every x ∈ B 0 .
Proof. In the proof, we only consider the difficult case B 0 = X; the case B 0 = X is similar. Let u : X → R be Lipschitz, with a constant κ > 0. Write g = C(1, c µ )M ♯,p 1,B 0 u, where the constant C(1, c µ ) > 0 is as in the proof of Lemma 4.12. First we show that 4g| B 0 is a p-weak upper gradient of u| B 0 with respect to B 0 .
To begin with, we observe that {y ∈ B 0 : g(y) > λ} is an open set if λ ∈ R. Hence, the function g| B 0 is Borel in B 0 . Fix a curve γ : [0, ℓ γ ] → B 0 ⊂ X, and then fix a natural number n ≥ 2 satisfying condition At the end, we will let n tend to infinity. We consider the covering [0, Consider a fixed i = 0, . . . , n − 2. Note first that Moreover, since B 0 X, we can choose a Whitney ball Q i ∈ W(B 0 ) such that x i ∈ Q i ; we refer to §4.2. By using inequalities (5.5) and (5.6), it is straightforward to show that x i , x i+1 ∈ Q * i . Hence, proceeding as in the proof of Lemma 4.12, we see that Thus, we obtain that By taking n → ∞ and using the continuity of u, together with the facts that x 0 (n) → γ(0) and x n−1 (n) → γ(ℓ γ ), we conclude that |u(γ(0)) − u(γ(ℓ γ ))| ≤ γ 4g ds .
This inequality shows that 4g| B 0 is a p-weak upper gradient of u| B 0 with respect to B 0 , and since 0 ≤ 4g ≤ 4C(1, c µ )κ, it also holds that 4g| B 0 ∈ L p (B 0 ). Inequality (5.4) for C(c µ ) = 4C(1, c µ ) now follows from the fact that the restriction g u | B 0 is the minimal p-weak upper gradient of u| B 0 with respect to the (open) ball B 0 ; we refer to [1,Lemma 2.23].
The following result is now a consequence of our main result, Theorem 4.3. This result can be further strenghtened by using Theorem 3.6, but we leave details to the interested reader.
Theorem 5.7. Suppose that X is a geodesic space and fix 1 < p < ∞. Suppose that there are constants K > 0 and τ ≥ 1 such that the (1, p)-Poincaré inequality holds whenever B is a ball in X and g ∈ L p loc (X) is a p-weak upper gradient of u ∈ Lip(X). Then there exists a number 0 < ε < p − 1 and a constant C > 0, both of which are quantitative, such that inequality holds whenever B ⊂ X is a ball, u ∈ Lip(X) and g ∈ L p loc (X) is a p-weak upper gradient of u. Proof. From the above considerations and the standing assumptions, it follows that the family {D 1,p N (u) : u ∈ Lip(X)} is a D-structure in X, with exponents p and β = 1, and constants K > 0 and τ ≥ 1. This allows us to fix 0 < ε < ε 0 as in Corollary 4.5. Fix also a ball B ⊂ X, and write B 0 = 2B and B 0 = {B(x, r) : B(x, 2r) ⊂ B 0 }. Since B ∈ B 0 , we have We apply Corollary 4.5 with the ball B 0 ⊂ X and with the minimal p-weak upper gradient g u ∈ L p loc (X) of u. Lemma 5.3 is needed to obtain the estimate for the right-hand side of (4.6). At the end we use the fact that g p−ε u ≤ g p−ε almost everywhere if g ∈ L p loc (X) is any p-weak upper gradient of u. Along the same lines, we can also prove a version of the Keith-Zhong theorem for general D-structures in geodesic spaces. This result is stated in Theorem 5.8 below. This result is a true generalization of Theorem 5.7 but it is unknown to the authors whether the additional minimality condition in the statement below can be removed.
Theorem 5.8. Suppose that we are given a D-structure in a geodesic space X, with exponents 1 < p < ∞ and 0 < β ≤ 1, and constants K > 0 and τ ≥ 1. Fix η > 0. Then there exists a number 0 < ε < p − 1 and a constant C > 0, both of which are quantitative, such that inequality holds whenever B ⊂ X is a ball, u ∈ Lip β (X) and g ∈ D(u) satisfies the following minimality condition: g ≤ ηM ♯,p β,B 0 u almost everywhere in B 0 = 2B. Theorem 5.8 can also be further strenghtened by using Theorem 3.6, but again we leave details to the interested reader.

Axiomatic Sobolev spaces
A given D-structure gives rise to a Sobolev space; cf. [5]. Our main result in this section is a certain norm-equivalence for such spaces, Theorem 6.2.

Sobolev spaces and D-structures
Let us begin with the definition of an abstract Sobolev space that is defined in terms of a Dstructure. Our treatment is inspired by [5]. See also [6,15] for further references on this type of abstract Sobolev spaces. Definition 6.1. Given a D-structure D in X, with exponents 1 ≤ p < ∞ and 0 < β ≤ 1, the associated Sobolev space W p β (X, D) is the completion [13] of the vector space {u ∈ Lip β (X) : u W p β (X,D) < ∞} that is equipped with the norm 3 In order to formulate our results, we need a global version of the maximal function (4.1). To this end, we write B = {B : B ⊂ X is a ball } and denote x ∈ X , whenever u : X → R is a β-Hölder function, i.e., u ∈ Lip β (X). By applying this global maximal function in Theorem 6.2 below, we provide a structure independent representation for the Sobolev norm that arises from an appropriate D-structure; more specifically, we need to additionally assume that ηM ♯,p β u ∈ D(u) whenever u is a β-Hölder function on X. Here η is a constant that is independent of u. As we will see, this assumption holds in various applications. Theorem 6.2. Suppose we are given a D-structure D in a geodesic space X, with exponents 1 < p < ∞ and 0 < β ≤ 1. Let η > 0 and suppose that ηM ♯,p β u ∈ D(u) for every u ∈ Lip β (X). Then there exists a constant C = C(K p,p , β, p, c µ , η) ≥ 1 such that C −1 u W p β (X,D) ≤ u p L p (X) + M ♯,p β u p L p (X) 1/p ≤ C u W p β (X,D) (6.3) whenever u ∈ Lip β (X).
Remark 6.4. Inequality (6.3) holds also when either one of the two quantities u W p β (X,D) , u p L p (X) + M ♯,p β u p L p (X) 1/p is infinite. In this case we can conclude that actually both of these quantities are infinite.
Proof of Theorem 6.2. The left inequality in (6.3) follows from the definitions and the assumption that ηM ♯,p β u ∈ D(u) for every u ∈ Lip β (X). To prove the right inequality, we fix a point x 0 ∈ X and denote B j = B(x 0 , j) and B j = {B = B(x, r) : 2B ⊂ B j } for j ∈ N.
Fix u ∈ Lip β (X) and g ∈ D(u). Observe that The following corollary is a universality result for Haj lasz-Sobolev spaces M β,p (X). Namely, any abstract Sobolev space, rising from a suitable D-structure, turns to be isomorphic to this particular Sobolev space M β,p (X). Corollary 6.7. Suppose we are given a D-structure D A in a geodesic space X, with exponents 1 < p < ∞ and 0 < β ≤ 1. Assume that there exists η > 0 such that ηM ♯,p β u ∈ D A (u) for every u ∈ Lip β (X). Let W β p (X) = W β p (X, D A ). Then there exists C ≥ 1 such that whenever u ∈ Lip β (X). Moreover, there exists a unique Banach-space isomorphism between the spaces W p β (X) and M β,p (X) which is the identity on W p β (X) ∩ Lip β (X) = M β,p (X) ∩ Lip β (X).
Applying Theorem 6.2 with the two D-structures D A and D β,p H yields the claim. Indeed, recall that by definitions W β p (X) = W β p (X, D A ) and M β,p (X) = W p β (X, D β,p H ).

Universality of Newtonian spaces
Fix u ∈ Lip(X) and 1 < p < ∞. Recall from §5 that D 1,p N (u) is the family of all p-weak upper gradients g ∈ L p loc (X) of the function u. Arguing as in §5, we find that D 1,p N = {D 1,p N (u) : u ∈ Lip(X)} is a D-structure with exponents β = 1 and p, if we assume that the (1, p)-Poincaré inequality condition (D1) holds. The associated abstract Sobolev space is the so-called Newtonian space N 1,p (X) = W p 1 (X, D 1,p N ) . We remark that this notation is not entirely standard since Lipschitz functions need not be dense when the more conventional approach [14], [1,Definition 1.17] to the Newtonian space is adopted. However, when X supports a (1, p)-Poincaré inequality (6.10) for all measurable functions instead of Lipschitz functions only, then the density result holds and the two definitions give isomorphic Banach spaces; c.f. [1, Theorem 5.1].
The following corollary extends and complements [14,Theorem 4.9] and [15,Theorem 4.3]. Observe that, by Corollary 6.7 and transitivity, it also produces a universality result for Newtonian spaces N 1,p (X). Corollary 6.9. Suppose that X is a geodesic space and fix 1 < p < ∞. Suppose that there are constants K > 0 and τ ≥ 1 such that the (1, p)-Poincaré inequality holds whenever B is a ball in X and g ∈ L p loc (X) is a p-weak upper gradient of u ∈ Lip(X). Then there exists a constant C ≥ 1 such that C −1 u N 1,p (X) ≤ u M 1,p (X) ≤ C u N 1,p (X) whenever u ∈ Lip(X). Moreover, there exists a unique Banach-space isomorphism between the spaces M 1,p (X) and N 1,p (X) which is the identity on N 1,p (X) ∩ Lip(X) = M 1,p (X) ∩ Lip(X).
Proof. Arguing as in §5, we obtain a constant C(1, c µ ) > 0 such that 4C(1, c µ )M ♯,p 1 u ∈ L p loc (X) is a p-weak upper gradient of any given u ∈ Lip(X). The claim follows from Corollary 6.7.