In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥brα for r∈(0,1] and any point x from metric space.


Introduction
Let be an open subset of the Euclidean space R n . If the boundary of is sufficiently regular and 1 ≤ p < n, then the Sobolev embedding W 1,p ( ) → L p * ( ) holds, where p * := np n−p (see e.g. [1]). On the other hand, it was shown by Hajłasz-Koskela-Tuominen [7] that if W 1,p ( ) → L p * ( ), then satisifies the so-called measure density condition, i.e. there exists a constant c > such that for all x ∈ and all 0 < r ≤ 1 |B(x, r) ∩ | ≥ cr n . ( More recently, the result of Hajłasz-Koskela-Tuominen has been extended to the Slobodeckij-Sobolev spaces W s,p (see Zhou [12]). Namely, if W s,p ( ) → L np/(n−sp) ( ) for some s ∈ (0, 1) and p ≥ 1 such that sp < n, then satisfies (1).
The main objective of the paper is to study the relation between Sobolev inequalities on metric measure spaces and lower bound for measure. In particular, we prove the following result. Suppose that (X, ρ, μ) is a metric measure space equipped with doubling measure and let M 1,p (X) be the Hajłasz-Sobolev space. If M 1,p (X) → L q (X), where p < q, then there exists b > 0 such that for any x ∈ X and 0 < r ≤ 1, the following inequality holds μ(B(x, r)) ≥ br α , The remainder of the paper is structured as follows. In Section 2, we introduce the notations and recall the notion of Sobolev spaces on general metric measure spaces. Our principal assertion, concerning the sufficient and necessary conditions for Sobolev embeddings are formulated and proven in Section 3.

Preliminaries
Let (X, ρ, μ) be a metric measure space equipped with a metric ρ and the Borel regular measure μ. We assume throughout the paper that the measure of every open nonempty set is positive and that the measure of every bounded set is finite. Additionally, we assume that the measure μ satisfies a doubling condition. It means that, there exists a constant C d > 0 such that for every ball B(x, r), It is well known (see e.g. Lemma 8.1.13 in [9]) that the doubling condition implies that, there exists a positive constant D satisfying for all balls B(x 2 , r 2 ) and B(x 1 , r 1 ) with r 2 ≥ r 1 > 0 and x 1 ∈ B(x 2 , r 2 ). It follows from the above inequality that if X is bounded, then there exists b > 0 such that the following inequality holds for r < diamX On the other hand, if the metric measure space equipped with a doubling measure is not bounded, then inequality (2) does not necessarily hold. Furthermore, we need to recall the notion of Ahlfors regularity. We shall say that the metric measure space (X, ρ, μ) is Ahlfors s-regular if there exist constants b and B such that We are now in a position to recall the notion of Sobolev spaces on metric measure spaces (see also [4]). Let (X, ρ, μ) be a metric measure space. We say that a p-integrable function f belongs to the Hajłasz-Sobolev space M 1,p (X) if there exists non-negative g ∈ L p (X), called a generalized gradient, such that g(y)) a.e. for x, y ∈ X.
We equip the space M 1,p (X) with the norm where the infimum is taken over all the generalized gradients. Then M 1,p is a Banach space. For the basic properties of this kind of spaces, we refer to [2,[4][5][6][9][10][11].
Suppose that f is locally integrable and A is a measurable set of positive measure, then by f A we denote the integral average of the function f over the set A, i.e.,

Main Results
In this section, we show the sufficient and necessary conditions for Sobolev inequalities. We will start with the following proposition.
where p * = sp s−p . Moreover, there exists C = C(s, p, b), depending on s, p, b, such that for each u ∈ M 1,p (X), the following inequality holds Proof Taking σ = 2 in Theorem 8.7 from [5], we have where the constant C depends on p and s. Hence, we get Thus, by the Hölder inequality, we obtain Finally, by passing to diamX with r, we obtain the desired result.
Next, we state necessary conditions for Sobolev embeddings. The proof of the following theorem relies on the methods established by Carron (see [3] and proof of Lemma 2.2 in [8]).

Theorem 3.2 Suppose that (X, ρ, μ) is a metric measure space with the doubling measure.
If where q > p, then there exists b = b(p, q, C pq ) such that μ (B(x, r)) ≥ br α , for r ∈ (0, 1], where 1 p − 1 q = 1 α and C p,q is the constant of the embedding.
Remark 1 Let us stress that b does not depend on the doubling constant C d .
Proof For each u ∈ M 1,p (X) we have where g is a generalized gradient of u. For a fixed x ∈ X and R > 0, let us define a Lipschitz function u R as follows It is easily seen that as a generalized gradient we can take Since the support of u is contained in B(x, R), we get in view of the Hölder inequality Let us fix r ≤ 1 and x ∈ X. Then, μ(B(x, r)) ≥ 1 2C pq α r α or μ(B(x, r)) ≤ 1 2C pq α r α . In the first case we have the desired inequality. Thus we may assume that μ (B(x, r) Consequently, the structure of u δ and g δ implies the following inequality 1 (2C pq ) p μ(B(x, δ)) − p α ≤ .