Resistance conditions, Poincaré inequalities, the Lip-lip condition and Hardy’s inequalities

Abstract This note investigates weaker conditions than a Poincaré inequality in analysis on metric measure spaces. We discuss two resistance conditions which are stated in terms of capacities. We show that these conditions can be characterized by versions of Sobolev–Poincaré inequalities. As a consequence, we obtain so-called Lip-lip condition related to pointwise Lipschitz constants. Moreover, we show that the pointwise Hardy inequalities and uniform fatness conditions are equivalent under an appropriate resistance condition.


Introduction
Rather standard assumptions in analysis on a metric measure space pX, d, µq are that the measure is doubling and that the space supports a Poincaré inequality, see [3] and [5]. The space is said to support a weak p1, pq-Poincaré inequality with 1 ≤ p ă 8, if there exist c P ą 0 and σ ≥ 1 such that for any x P X and r ą 0, and for every locally integrable function f in X, ż Bpx,rq conditions imply, for example, the Sobolev embedding theorem, which is a central tool in analysis on metric measure spaces, see [3] and [5].
The goal of this note is to consider weaker conditions than the Poincaré inequality. More precisely, the weak p1, pq-Poincaré inequality implies socalled resistance condition 1 c R µpBpx, rqq r p ≤ cap Lip,p pBpx, rq, Bpx, 2rqq ≤ c R µpBpx, rqq r p , (1. 1) for every x P X and r ą 0 with a uniform constant c R ≥ 1. Here we consider the capacity defined as cap Lip,p pBpx, rq, Bpx, 2rqq " inf where the infimum is taken over all Lipschitz continuous functions f in X with f ≥ 1 in Bpx, rq and f " 0 in XzBpx, 2rq. The resistance condition is considerably weaker than the Poincaré inequality. Even in the case when the space is complete, the resistance condition does not imply quasiconvexity of the space and, as a consequence, it is not equivalent with the Poincaré inequality, see [9]. A similar condition has been previously employed, for example in [1] and [4] in connection with the Dirichlet forms on metric measure spaces.
Several versions of the resistance condition are available and it is not obvious which is the best approach. In this note, we discuss two conditions called the p-resistance conductor condition and the p-strong resistance conductor condition. These conditions seem to be stronger than (1.1) and, as we shall see, they can be characterized by versions of Sobolev-Poincaré inequalities in the same way as in [11]. For results in metric measure spaces, see also [8]. Using the results of [7], we conclude that if X is a metric space with µ doubling and that satisfies a p-strong resistance conductor condition, then so-called Lip-lip condition related to pointwise Lipschitz constants holds true. Moreover, we show that the pointwise Hardy inequalities and uniform fatness conditions are equivalent in our context. This is closely related to results of [10] .

Preliminaries
From now on, let pX, d, µq be a metric measure space. Here µ is a doubling measure, that is, there exists c D ≥ 1, called the doubling constant of µ, such that for all x P X and r ą 0, µpBpx, 2rqq ≤ c D µpBpx, rqq, where Bpx, rq " ty P X : dpx, yq ă ru. for every x, y P X. In this case the Lipchitz constant of f is defined to be the infimum over all constants c ą 0 for which (2.1) holds and LippXq denotes the class of Lipschitz functions on X.
We denote where x P X. If f P LippXq, then for every x P X the lower local Lipschitz constant of u is defined by

Sobolev-Poincaré inequalities and resistance conditions
Let 1 ≤ p ă 8 and pE, Gq be a pair of sets in X, where E is a µ-measurable subset of an open set G. We define the capacity of pE, Gq in X as where the infimum is taken over all f P LippXq with f ≥ 1 in E and f " 0 in XzG.
Let Ω be an open and bounded subset of X. We want to consider Poincaré inequalities for functions that are not necessary zero on the boundary of the domain. To this end, we shall need the concept of conductivity. Let G be an open subset of Ω and E Ă G a µ-measurable set. Then where the infimum is taken over all f P LippΩq with f ≥ 1 in E and f " 0 in ΩzG. We also denote cap Lip,p pE, G; Xq " cap Lip,p pE, Gq.
Definition 3.1. The space X satisfies the p-resistance conductor condition, if there exists c R ≥ 1 such that for any x P X, 0 ă r ă diampXq{2 and E Ĺ Bpx, rq, we have 1 c R µpEq r p ≤ cap Lip,p pE, Bpx, rqq. The following capacitary strong type estimate will be useful later.
Proof. If E t " ∅ for every t ą 0, then there is nothing to prove. Hence, we may assume that E t ‰ ∅ for some t ą 0. In this case Note that E t Ĺ Bpx, rq for every t ą 0. We define f j by Hence, by (3.4) and (3.5), we arrive at The next result shows that the p-resistance conductor condition can be characterized by a Sobolev type inequality for functions vanishing on a relatively large set.
Theorem 3.6. The space X satisfies the p-resistance conductor condition if and only if for any x P X, 0 ă r ă diampXq{2 and f P LippBpx, 2rqq, for Proof. If E t " tz P Bpx, rq : |f pzq| ≥ tu " ∅ for every t ą 0, then the inequality is trivial. Assume then that there exists t ą 0 such that ∅ ‰ E t Ĺ Bpx, rq. The Cavalieri principle and the previous lemma imply that Conversely, let x P X, 0 ă r ă diampXq{2 and E Ĺ Bpx, rq. For any f P LippBpx, 2rqq such that f " 0 in Bpx, 2rqzBpx, rq and f ≥ 1 in E, we have that Raising both sides to the power p and taking infimum over all such functions, we arrive at cap Lip,p pE, Bpx, rqq ≥ µpEq cr p . The proof of the following capacitary strong type estimate is similar to Lemma 3.3.
Next, we introduce another resistance condition. We obtain a similar characterization of the p-strong resistance conductor condition as in Theorem 3.6.
Theorem 3.9. The space X satisfies the p-strong resistance conductor condition if and only if, for any x P X, 0 ă r ă diampXq{2 and f P LippBpx, 2rqq with f " 0 in Bpx, 2rqzG for G is an open set in Bpx, 2rq, we haveˆż Proof. Let us start with the sufficiency. If f " 0 in Bpx, rq, there is nothing to prove. Hence, we may assume that E t " tz P Bpx, rq : |f pzq| ≥ tu ‰ ∅ for some t ą 0. If G Ď Bpx, rq, then the result follows from Theorem 3.6. If not, G X pBpx, 2rqzBpx, rqq ‰ ∅ and by the p-strong resistance conductor condition, µpE t q ≤ c R r p cap Lip,p pE t , G; Bpx, 2rqq.
Hence, by the Cavalieri principle and Lemma 3.7, we havê Conversely, let x P X, 0 ă r ă diampXq{2, E Ĺ Bpx, rq and G be an open set such that E Ă G Ĺ Bpx, 2rq and G X pBpx, 2rqzBpx, rqq ‰ ∅. Given δ ą 0, there exists f P LippBpx, 2rqq such that f " 0 in Bpx, 2rqzG, f ≥ where the infimum is taken over all open G that E Ă G Ĺ Bpx, 2rq, G X pBpx, 2rqzBpx, rqq ‰ ∅. Moreover, taking G " Bpx, rq, by Theorem 3.6, the space X satisfies the p-resistance conductor condition. Therefore, the fact that X satisfies this condition, together with (3.10), implies that the space X satisfies the p-strong resistance conductor condition.
Theorem 3.11. If X satisfies the p-strong resistance conductor condition, then for any f P LippBpx, 2rqq, x P X and 0 ă r ă diampXq{2, we havê where c depends only on p and c R .
Corollary 3.14. Let f P LippBpx, 2rqq, x P X and 0 ă r ă diampXq{2. If X satisfies the p-strong resistance conductor condition, then where c depends only on p, c D and c R .
Proof. It follows from Theorem 3.11 and Hölder's inequality.
To finish this section let us observe that the 1-strong resistance conductor condition implies, by Corollary 3.14 and Hölder's inequality, the p1, pq-Poincaré inequality for any locally Lipschitz function. That is, X satisfies a p1, pq-Poincaré inequality for any locally Lipschitz function in the sense of [3,Chapter 4]. Therefore, in a similar way as in [3,Corollary 4.19], it follows that X satisfies the pp, pq-Poincaré inequality for any locally Lipschitz function. Finally, as in [6], we can see that X satisfies the p-strong resistance condition.

Lip-lip condition, p-fatness and Hardy inequalities
It is shown in [7] that the Poincaré inequality implies the Lip-lip condition when the space is complete. In this section, we show first that the Lip-lip condition follows when the space satisfies a p-strong resistance conductor condition even without completeness. Our argument is similar to the one used in [7,Section 4.3] with minor changes. Let x P X. Since µ is doubling, there exists a measurable set A in X containing x where Lipf is a continuous function. Moreover, by the Lebesgue differentiation theorem (see [5,Theorem 1.8]  The proof of the next lemma follows from a straight forward adaption the argument in [7,Section 4.3] applying (4.1) and Remark 2.4.
As in [7,Section 4.3] applying Corollary 3.14 we obtain the following result.
Theorem 4.3. If X satisfies the p-strong resistance conductor condition, then for any x P X and f P LippXq, where c depends only on p, c D and c R .
Next we recall two definitions.  " c H`MLd Ω pxq plip uq p pxq˘1 {p holds for almost every x P Ω. Here d Ω pxq " dpx, XzΩq and M Ld Ω pxq denotes the Hardy-Littlewood maximal function with the restricted radii.
The following result is a modification of the corresponding result for spaces satisfying a Poincaré inequality, see [10]. It is clear that f is a 1{r-Lipschitz function such that f " 1 in Bpx, rq, 0 ≤ f ≤ 1 and f " 0 in XzBpx, 2rq. We may use f as an admissible function in the definition of the capacity and obtain Therefore, it is enough to show that there exists c ą 0 such that µpBpx, rqq plip f q p dµ, (4.9) for any f P LippXq such that f " 0 in XzBpx, 2rq and f ≥ 1 in Bpx, rq X pXzΩq.
Let l " p2pL`1qq´1, where L is the constant in the pointwise p-Hardy inequality. The doubling property implies that c D µpBpx, lrqq ≥ l s µpBpx, rqq. First let us assume that f B ą l s {p2c D q, where B " Bpx, rq. Since f P LippBpx, 4rqq and f " 0 in Bpx, 4rqzBpx, 2rq, Theorem 3.6 and Hölder's inequality imply that l s 2c D ă f B ≤ c D µpBpx, 2rqq Hence (4.9) holds in that case. On the other hand, if f B ≤ l s {p2c D q, then we can argue as in [10] and obtain (4.9) also in that case.
Recall that Corollary 3.14 states that under the p-strong resistance conductor condition, we have for every x P X. (iv) Ω admits the pointwise p-Hardy's inequality.