Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces
aa r X i v : . [ m a t h . M G ] J a n DOUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZEDMEASURES ON GROMOV HYPERBOLIC SPACES
CLARK BUTLER
Abstract.
We generalize the recent results of Bj¨orn-Bj¨orn-Shanmugalingam [2] con-cerning how measures transform under the uniformization procedure of Bonk-Heinonen-Koskela for Gromov hyperbolic spaces [4] by showing that these results also hold inthe setting of uniformizing Gromov hyperbolic spaces by Busemann functions that weintroduced in [9]. In particular uniformly local doubling and uniformly local Poincar´einequalities for the starting measure transform into global doubling and global Poincar´einequalities for the uniformized measure. We then show in the setting of uniformizationsof universal covers of closed negatively curved Riemannian manifolds equipped with theRiemannian measure that one can obtain sharp ranges of exponents for the uniformizedmeasure to be doubling and satisfy a 1-Poincar´e inequality. Lastly we introduce the pro-cedure of uniform inversion for uniform metric spaces, and show that both the doublingproperty and the p -Poincar´e inequality are preserved by uniform inversion for any p ≥ Introduction
Broadly speaking our work in this paper has three closely related objectives. The primaryobjective is to generalize the recent results of Bj¨orn-Bj¨orn-Shanmugalingam [2] concerninghow measures transform under the uniformization procedure of Bonk-Heinonen-Koskela forGromov hyperbolic spaces [4]. We will consider how measures transform under the gen-eralization of this uniformization procedure that we introduced in [9]. As in [2], we willshow that uniformizing these measures upgrades uniformly local doubling properties anduniformly local Poincar´e inequalities to global doubling and global Poincar´e inequalities forthe uniformized space. Our results allow us to construct a number of interesting new un-bounded metric measure spaces supporting Poincar´e inequalities. A particularly importantexample is uniformizations of hyperbolic fillings of unbounded metric spaces, which play akey role in our followup work [7] concerning extension and trace theorems for Besov spaceson noncompact doubling metric measure spaces.Our second objective is to show that in the presence of a cocompact isometric discretegroup action on the Gromov hyperbolic space we start with, it is often possible to apply thetheorems of [2] to a much wider range of exponents than the ones considered there, leading inseveral cases to ranges that we can verify are sharp due to well-known results on Patterson-Sullivan measures in these contexts. In Theorem 1.4 we tie this threshold to the volumegrowth entropy of universal covers of closed negatively curved Riemannian manifolds. InRemark 3.10 we briefly explain how Patterson-Sullivan measures on the Gromov boundaryarise as renormalized limits of the uniformized measures considered here and in [2].The final topic that we consider is procedures for transforming metric measure spaces thatpreserve the doubling property and p -Poincar´e inequalities for a given p ≥
1. Oftentimesin analysis on metric spaces it is preferable to work on either a bounded or an unboundedspace depending on the nature of the question under consideration. Thus there has beena significant amount of interest in procedures for passing back and forth between bounded and unbounded spaces while retaining as much information as possible. In the abstractmetric space setting transformations between bounded and unbounded spaces can be realizedthrough inversions [6], which generalize the classical notion of M¨obius inversions in thecomplex plane. It was shown by Li-Shanmugalingam [16] that measures can be transformedunder these inversions in such a way that a number of desirable properties can be preserved.However they were not able to obtain unconditional invariance of Poincar´e inequalities underinversions, and they in fact showed that Poincar´e inequalities cannot always be preserved[16, Example 3.3.13]. In the final section of this paper we introduce an alternative inversionoperation based on uniformization that is specialized to uniform metric spaces. With theadditional assistance of some results from [2] we will show that this operation preservesPoincar´e inequalities. A more in-depth discussion of this is given in Section 5.We now introduce the setting of [9] and [2] in order to state our main theorems. Forprecise definitions regarding general notions in Gromov hyperbolic spaces we refer to Section ?? , while for a more detailed treatment of the uniformization procedure discussed in thisintroduction we refer to Section 2. We begin with a proper geodesic δ -hyperbolic metricspace ( X, d ), meaning that geodesic triangles in X are δ -thin for a constant δ ≥
0. The
Gromov boundary ∂X of X is the collection of all geodesic rays γ : [0 , ∞ ) → X up to theequivalence relation that two geodesic rays are equivalent if they are at bounded distancefrom one another. For a given geodesic ray γ : [0 , ∞ ) → X , the Busemann function b γ : X → R associated to γ is defined by the limit(1.1) b γ ( x ) = lim t →∞ d ( γ ( t ) , x ) − t. We then define(1.2) B ( X ) = { b γ + s : γ a geodesic ray in X , s ∈ R } , and refer to any function b ∈ B ( X ) as a Busemann function on X . See [9, (1.4-1.5)] forfurther details. The Busemann functions b ∈ B ( X ) are all 1-Lipschitz functions on X . For aBusemann function b of the form b = b γ + s for some s ∈ R , we define the endpoint ω ∈ ∂X of γ to be the basepoint of b and say that b is based at ω .For z ∈ X we define b z ( x ) = d ( x, z ) to be the distance from z . We augment the set ofBusemann functions with the set of translates of distance functions on X ,(1.3) D ( X ) = { b z + s : z ∈ X, s ∈ R } . For b ∈ D ( X ) with b = b z + s for some z ∈ X and s ∈ R we then refer to z as the basepoint of b , in analogy to the case of Busemann functions. We write ˆ B ( X ) = D ( X ) ∪ B ( X ). Thenall functions b ∈ ˆ B ( X ) are 1-Lipschitz.For each b ∈ ˆ B ( X ) and each ε > ρ ε,b on X by ρ ε,b ( x ) = e − εb ( x ) . For a curve γ in X we let ℓ ε,b ( γ ) = Z γ ρ ε,b ds, denote the line integral of ρ ε,b along γ . We let ( X ε,b , d ε,b ) denote the metric space obtainedby conformally deforming X by the density ρ ε,b , i.e., defining the new distance d ε,b for x, y ∈ X by(1.4) d ε,b ( x, y ) = inf ℓ ε,b ( γ ) , with the infimum taken over all curves γ joining x to y . The metric sapce X ε,b is boundedif and only if b ∈ D ( X ) [9, Proposition 4.4]. OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 3
Our main results concern the corresponding effect of this conformal deformation on mea-sures on X . We will require the following key definition. For the entirety of this paper a metric measure space is a triple ( X, d, µ ) consisting of a metric space (
X, d ) equipped witha Borel measure µ . Definition 1.1.
Let (
X, d, µ ) be a metric measure space and let B X ( x, r ) denote the openball of radius r > x ∈ X . The measure µ is doubling if there is a constant C µ ≥ x ∈ X and r > µ ( B X ( x, r )) ≤ C µ µ ( B X ( x, r )) . If the inequality (1.5) only holds for balls of radius at most R then we will say that µ is doubling on balls of radius at most R . We will alternatively say that µ is uniformly locallydoubling if there is an R > µ is doubling on balls of radius at most R .We let µ be a given Borel measure on X that is doubling on balls of radius at most R with constant C µ . For each β > µ β,b on X by(1.6) dµ β,b ( x ) = ρ β,b ( x ) dµ ( x ) = e − βb ( x ) dµ ( x ) , for x ∈ X . We will consider µ β,b as a measure on X ε,b and extend it to the completion ¯ X ε,b by setting µ β,b ( ∂X ε,b ) = 0, where ∂X ε,b = ¯ X ε,b \ X ε,b denotes the complement of X ε,b insideits completion. Our first theorem shows that there is a threshold β depending only on R and C µ such that if β ≥ β then the uniformly locally doubling measure µ on X transformsinto a measure µ β,b on ¯ X ε,b that is doubling at all scales.Our theorem requires two additional hypotheses on X and the density ρ ε,b , which webriefly summarize here. We recall that we are assuming that X is a proper geodesic δ -hyperbolic space and that b ∈ ˆ B ( X ). The first is that X is K -roughly starlike from thebasepoint ω ∈ X ∪ ∂X of the chosen function b ∈ ˆ B ( X ) for a given constant K ≥ x ∈ X lies within distance K of ageodesic ray or line starting from ω . We defer a precise definition to Section ?? as one mustdistinguish the cases ω ∈ X and ω ∈ ∂X .The second requirement is that ρ ε,b is a Gehring-Hayman density for X with constant M ≥ GH-density ). This means that for each x, y ∈ X and each geodesic γ joining x to y we have(1.7) d ε,b ( x, y ) ≤ M ℓ ε,b ( γ ) . In other words, ρ ε,b is a GH-density if geodesics in X minimize distance in X ε,b up to auniversal multiplicative constant. This requirement is not as stringent as it first appears; bythe work of Bonk-Heinonen-Koskela [4, Theorem 5.1] there is an ε = ε ( δ ) > δ such that ρ ε,b is a GH-density with constant M = 20 for any b ∈ ˆ B ( X ) and any0 < ε ≤ ε . For CAT( −
1) spaces one may use a threshold ε = 1 instead [9, Theorem 1.10].The rough starlikeness hypothesis and the GH-density hypothesis together guarantee thatthe conformal deformation X ε,b has a number of nice properties by our previous results in[9] that are summarized in Section 2 and are used heavily in the proofs of our theorems.We then have the following theorem; below “the data” refers to the collection of param-eters δ , K , ε , M , β , R , and C µ . Theorem 1.2.
There is β = β ( R , C µ ) > such that if β ≥ β then the measure µ β,b on ¯ X ε,b is doubling with constant C µ β depending only on the data. This theorem generalizes the the main result of Bj¨orn-Bj¨orn-Shanmugalingam [2, Theo-rem 1.1] to the setting in which the uniformization X ε,b is potentially unbounded, i.e., the CLARK BUTLER case b ∈ B ( X ). The case b ∈ D ( X ) is more or less already contained in [2, Theorem 1.1],with the exception that they only consider the original parameter range 0 < ε ≤ ε of Bonk-Heinonen-Koskela. An explicit value β =
17 log C µ R is given in [2, Theorem 1.1]; a similarexplicit estimate for β can be extracted from our proof. In the process of proving Theorem1.2 we formulate a useful criterion (Proposition 3.3) for checking that µ β,b is doubling on¯ X ε,b . This criterion will be used to verify Theorem 1.4 below as well as some key claims inour followup work [7].The doubling property for µ β,b on ¯ X ε,b is the key property needed to transform uniformlylocal p -Poincar´e inequalities on X into global p -Poincar´e inequalities on X ε,b . The followingtheorem makes this claim precise. We refer to Section 4 for the precise definitions of uni-formly local p -Poincar´e inequalities and global p -Poincar´e inequalities. We retain the samehypotheses regarding rough starlikeness and the GH-density property that we assumed inTheorem 1.2. Below “the data” refers to the parameters δ , K , ε , M , β , the doubling con-stant C µ β,b for µ β,b on ¯ X ε,b , the power p , the radius and the constants C PI and λ appearingin the uniformly local p -Poincar´e inequality (4.2) as well as the local doubling radius andconstant C µ for µ . Theorem 1.3.
Suppose that the metric measure space ( X, d, µ ) is uniformly locally doublingand supports a uniformly local p -Poincar´e inequality for some p ≥ . Suppose further thatfor a given β > we have that µ β,b is doubling on ¯ X ε,b with constant C µ β . Then the metricmeasure spaces ( X ε,b , d ε,b , µ β,b ) and ( ¯ X ε,b , d ε,b , µ β,b ) each support a p -Poincar´e inequalitywith constant C ∗ PI depending only on the data. By combining Theorems 1.2 and 1.3, we see that if we assume (
X, d, µ ) is uniformlylocally doubling and supports a uniformly local p -Poincar´e inequality then for β ≥ β wealways have that ( X ε,b , d ε,b , µ β,b ) is doubling and supports a p -Poincar´e inequality, and thesame is true with ¯ X ε,b replacing X ε,b . For b ∈ D ( X ) Theorem 1.3 essentially follows directlyfrom [2, Theorem 1.1] and its proof. For Busemann functions b ∈ B ( X ) our uniformizationconstruction in [9] is designed such that minimal modifications to the proofs in [2] arerequired. We emphasize that Theorem 1.3 does not require us to restrict to the range β ≥ β considered in Theorem 3.3; it only requires that µ β,b is doubling on ¯ X ε,b .As indicated previously, when X comes equipped with a cocompact discrete isometricgroup action it is possible to significantly improve Theorem 1.2 by obtaining a better, oftensharp threshold β for µ β,b to be doubling. This is the content of Theorem 3.9 in Section 3.We highlight here an interesting corollary of this theorem that illustrates the power of thismethod.We consider a complete simply connected n -dimensional Riemannian manifold X withsectional curvatures ≤ − X . We let µ denote the Riemannian volume on X , which is Γ-invariant. The volumegrowth entropy of X is given by the limit for any x ∈ X ,(1.8) h X = lim R →∞ log µ ( B X ( x, R )) R .
For the existence of this limit see [17]. The quantity h X shows up in many places, forinstance it is also equal to the topological entropy of the geodesic flow on the unit tangentbundle of the quotient of X by Γ [17]. The constants in Theorem 1.4 are uniform in thesense that they do not depend on the choice of function b ∈ ˆ B ( X ). Theorem 1.4.
For each β > h X the metric measure spaces ( X ,b , d ,b , µ β,b ) and ( ¯ X ,b , d ,b , µ β,b ) for b ∈ ˆ B ( X ) are doubling and support a -Poincar´e inequality with uniform constants. OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 5
In Remark 3.10 we explain why the threshold h X is sharp. The constants in Theorem1.4 are uniform in the sense that they do not depend on the choice of b ∈ ˆ B ( X ), althoughthey will depend on the choice of exponent β > h X .Our results regarding preservation of Poincar´e inequalities for uniform metric spaces areproved in Section 5. These results follow formally by combining Theorems 1.2 and 1.3above with results in [8] and [2]; since the initial setup is quite different from that of ourtheorems above we have isolated the discussion of those results to Section 5. The rest of thepaper is structured as follows: in Section 2 we review some results from our previous work[9] regarding uniformizing Gromov hyperbolic spaces and extend some results from [2] tothe setting of uniformizing by Busemann functions. In Section 3 we analyze the doublingproperties of the uniformized measure (1.6) and prove Theorem 1.2. Lastly in Section 4 weprove Theorems 1.3 and 1.4. 2. Uniformization
Definitions.
Let X be a set and let f , g be real-valued functions defined on X . For c ≥ f . = c g if | f ( x ) − g ( x ) | ≤ c, for all x ∈ X . If the exact value of the constant c is not important or implied by contextwe will often just write f . = g . The relation f . = g will sometimes be referred to as a roughequality between f and g . Similarly for C ≥ f, g : X → (0 , ∞ ), we willwrite f ≍ C g if for all x ∈ X , C − g ( x ) ≤ f ( x ) ≤ Cg ( x ) . We will write f ≍ g if the value of C is implied by context. We will write f . C g if f ( x ) ≤ Cg ( x ) for all x ∈ X and f & C g if f ( x ) ≥ C − g ( x ) for x ∈ X . Thus f ≍ C g ifand only if f . C g and f & C g . As with the other notation, we will drop the constant C and just write f . g or f & g if the value of C is implied by context. We will generallystick to the convention of using c ≥ C ≥ δ – the constants depend on we willwrite c = c ( δ ), etc. At the beginning of each section we will indicate on what parameters theimplied constants of the inequalities . and & , the comparisons ≍ , and the rough equalities . = are allowed to depend. We will often reiterate these conditions for emphasis.For a metric space ( X, d ) we will write B X ( x, r ) = { y ∈ X : d ( x, y ) < r } for the open ballof radius r > x ∈ X . We write ¯ B X ( x, r ) = { y ∈ X : d ( x, y ) ≤ r } for theclosed ball of radius r > x . We note that the inclusion B X ( x, r ) ⊂ ¯ B X ( x, r )of the closure of the open ball into the closed ball can be strict in general. By conventionall balls B ⊂ X are considered to have a fixed center and radius, even though it may bethe case that we have B X ( x, r ) = B X ( x ′ , r ′ ) as sets for some x = x ′ , r = r ′ . All balls B ⊂ X are also considered to be open balls unless otherwise specified. We will write r ( B ) for the radius of a ball B . For a ball B = B X ( x, r ) in X and a constant c > cB = B X ( x, cr ) for the corresponding ball with radius scaled by c . For a subset E ⊂ X we write diam( E ) = sup { d ( x, y ) : x, y ∈ E } for the diameter of E and writedist( x, E ) = inf { d ( x, y ) : y ∈ E } for the infimal distance of a point x ∈ X to E .Let f : ( X, d ) → ( X ′ , d ′ ) be a map between metric spaces. We say that f is isometric if d ′ ( f ( x ) , f ( y )) = d ( x, y ) for x , y ∈ X . We recall that a curve γ : I → X is a geodesic if it is an isometric mapping of the interval I ⊂ R into X . We say that X is geodesic ifany two points in X can be joined by a geodesic. A geodesic triangle ∆ in X consists ofthree points x, y, z ∈ X together with geodesics joining these points to one another. Writing CLARK BUTLER ∆ = γ ∪ γ ∪ γ as a union of its edges, we say that ∆ is δ -thin for a given δ ≥ p ∈ γ i , i = 1 , ,
3, there is a point q ∈ γ j with d ( p, q ) ≤ δ and i = j . A geodesicmetric space X is Gromov hyperbolic if there is a δ ≥ X are δ -thin; in this case we will also say that X is δ -hyperbolic . When considering Gromovhyperbolic spaces X we will usually use the generic distance notation | xy | := d ( x, y ) for thedistance between x and y in X and the generic notation xy for a geodesic connecting twopoints x, y ∈ X , even when this geodesic is not unique.A metric space ( X, d ) is proper if its closed balls are compact. The Gromov boundary ∂X of a proper geodesic δ -hyperbolic space X is defined to be the collection of all geodesic rays γ : [0 , ∞ ) → X up to the equivalence relation of two rays being equivalent if they are at abounded distance from one another. We will often refer to the point ω ∈ ∂X correspondingto a geodesic ray γ as the endpoint of γ . Using the Arzela-Ascoli theorem it is easy to see ina proper geodesic δ -hyperbolic space that for any points x, y ∈ X ∪ ∂X there is a geodesic γ joining x to y . We will continue to write xy for any such choice of geodesic joining x to y . We will allow our geodesic triangles ∆ to have vertices on ∂X , in which case we will stillwrite ∆ = xyz if ∆ has vertices x, y, z .As in our previous work [9], we will use the notation ∂X for the Gromov boundary of X even though it conflicts with the notation ∂ Ω = ¯Ω \ Ω for the metric boundary of a metricspace (Ω , d ) inside its completion ¯Ω. Since we always assume that X is proper we will alwayshave ¯ X = X , so the metric boundary of X will always be trivial. Thus there will be noambiguity in using ∂X for the Gromov boundary as well.For x, y, z ∈ X the Gromov product of x and y based at z is defined by(2.1) ( x | y ) z = 12 ( | xz | + | yz | − | xy | ) . We can also take the basepoint of the Gromov product to be any function b ∈ ˆ B ( X ). For b ∈ ˆ B ( X ) the Gromov product based at b is defined by(2.2) ( x | y ) b = 12 ( b ( x ) + b ( y ) − | xy | ) . For b ∈ D ( X ), b ( x ) = d ( x, z ) + s this reduces to the notion of Gromov product in (2.1), aswe have ( x | y ) b = ( x | y ) z + s .We now consider an incomplete metric space (Ω , d ) and write ∂ Ω = ¯Ω \ Ω for the metricboundary of Ω in its completion ¯Ω. We write d Ω ( x ) := dist( x, ∂ Ω) for the distance of a point x ∈ Ω to the boundary ∂ Ω. An important observation that we will use without comment isthat d Ω defines a 1-Lipschitz function on Ω, i.e., for x, y ∈ Ω we have | d Ω ( x ) − d Ω ( y ) | ≤ d ( x, y ) . For a curve γ : I → Ω we write ℓ ( γ ) for the length of γ and say that γ is rectifiable if ℓ ( γ ) < ∞ . For an interval I ⊂ R and t ∈ I we write I ≤ t = { s ∈ I : s ≤ t } and I ≥ t = { s ∈ I : s ≥ t } . For a rectifiable curve γ : I → Ω we write γ − , γ + ∈ ¯Ω for theendpoints of γ ; writing t − ∈ [ −∞ , ∞ ) and t + ∈ ( −∞ , ∞ ] for the endpoints of I , theseare defined by the limits γ ( t − ) = lim t → t − γ ( t ) and γ ( t + ) = lim t → t + γ ( t ) in ¯Ω which existbecause ℓ ( γ ) < ∞ . Definition 2.1.
For a constant A ≥ I ⊂ R , a curve γ : I → Ω is A -uniform if(2.3) ℓ ( γ ) ≤ Ad ( γ − , γ + ) , OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 7 and if for every t ∈ I we have(2.4) min { ℓ ( γ | I ≤ t ) , ℓ ( γ | I ≥ t ) } ≤ Ad Ω ( γ ( t )) . The metric space Ω is A -uniform if it is locally compact and if any two points in Ω can bejoined by an A -uniform curve.We extend Definition (2.1) to the case of non-rectifiable curves γ : I → Ω by replacing(2.3) with the condition that d ( γ ( s ) , γ ( t )) → ∞ as s → t − and t → t + . We keep therequirement (2.4) the same. Observe that with this extended definition the inequality (2.4)implies that an A -uniform curve γ is always locally rectifiable , meaning that each compactsubcurve of γ is rectifiable. We note that it is easily verified from the definitions that theproperty of a curve γ being A -uniform is independent of the choice of parametrization of γ .Now let X be a proper geodesic δ -hyperbolic space. We define X to be K -roughly starlike from a point z ∈ X if for each x ∈ X there is a geodesic ray γ : [0 , ∞ ) → X such thatdist( x, γ ) ≤ K . Similarly for ω ∈ ∂X we define X to be K -roughly starlike from ω if foreach x ∈ X there is a geodesic line γ : R → X with γ | ( −∞ , ∈ ω and dist( x, γ ) ≤ K . When ∂X contains at least two points K -rough starlikeness from any point x ∈ X ∪ ∂X implies K ′ -rough starlikeness from all points of X ∪ ∂X for a constant K ′ ≥ ω immediately implies that ∂ ω X = ∅ .We fix a function b ∈ ˆ B ( X ) with basepoint ω ∈ X ∪ ∂X and let ε > ρ ε ( x ) = e − εb ( x ) is a GH-density on X with constant M . Since b is 1-Lipschitz wehave the Harnack type inequality for x, y ∈ X ,(2.5) e − ε | xy | ≤ ρ ε ( x ) ρ ε ( y ) ≤ e ε | xy | . We write X ε = X ε,b for the conformal deformation of X with conformal factor ρ andwrite d ε = d ε,b for the resulting distance on X ε . We write ℓ ε ( γ ) := ℓ ε,b ( γ ) for the lengthsof curves measured in the metric d ε and ℓ ( γ ) for the lengths of curves measured in X . Theproperness of X implies that X ε is locally compact. By [9, Theorem 1.4] the metric space X ε is incomplete and geodesics in X are A -uniform curves in X ε . In particular the metricspace ( X ε , d ε ) is A -uniform. Furthermore the space X ε is bounded if and only if b ∈ D ( X ).The proof of [9, Theorem 1.4] shows that when b ∈ D ( X ) all geodesics in X have finitelength in X ε , while in the case b ∈ B ( X ) geodesics have finite length if and only if they donot have the basepoint ω of b as an endpoint. For x ∈ ¯ X ε we write B ε ( x, r ) for the openball of radius r > x in the metric d ε on ¯ X ε , and for x ∈ X we write B X ( x, r )for the open ball of radius r centered at x in X .For x ∈ X ε write d ε ( x ) = d X ε ( x ) for the distance to the metric boundary ∂X ε of X ε .By [9, Theorem 1.6] there is a canonical identification ϕ ε : ∂ ω X → ∂X ε of the Gromovboundary of X relative to ω and the metric boundary ∂X ε of X ε ; we recall that ∂ ω X = ∂X if ω ∈ X and ∂ ω X = ∂X \{ ω } if ω ∈ ∂X . The correspondence is given by showing that anysequence { x n } in X converging to a point ξ ∈ ∂ ω X is a Cauchy sequence in X ε convergingto a point of ∂X ε .The local compactness of X ε implies by the Arzela-Ascoli theorem that, for a given x, y ∈ X , a minimizing curve γ for the right side of (1.4) always exists. It is easy to see thatsuch a curve must be a geodesic in X ε , from which we conclude that X ε is always geodesic.By [4, Proposition 2.20] the completion ¯ X ε of X ε is proper, and in particular is also locallycompact. A second application of Arzela-Ascoli then shows that ¯ X ε is also geodesic.We collect here two important quantitative results regarding the uniformization X ε fromour previous work [9]. The standing assumptions for the rest of this section are that X is a CLARK BUTLER proper geodesic δ -hyperbolic space with a given b ∈ ˆ B ( X ) such that X is K -roughly starlikefrom the basepoint ω of b , and that for a given ε > ρ ε ( x ) = e − εb ( x ) on X is aGH-density with constant M . All implied constants will depend only on δ , K , ε , and M . Lemma 2.2. [9, Lemma 4.7]
For x, y ∈ X we have (2.6) d ε ( x, y ) ≍ e − ε ( x | y ) b min { , | xy |} . Lemma 2.3. [9, Lemma 4.15]
For x ∈ X we have (2.7) d ε ( x ) ≍ ρ ε ( x ) . Lemmas 2.2 and 2.3 are stated for b ∈ B ( X ) in [9], however as noted in [9, Remark 4.24]the estimates for b ∈ D ( X ) can be deduced from the estimates for b ∈ B ( X ) by attaching aray to X at the basepoint of a given b ∈ D ( X ).We conclude this section by adapting two key claims from [2] to our setting. The firstclaim adapts [2, Theorem 2.10]. The proof is essentially the same. Lemma 2.4.
There is a constant C ∗ = C ∗ ( δ, K, ε, M ) ≥ such that for any x ∈ X andany < r ≤ d ε ( x ) we have the inclusions, (2.8) B X (cid:18) x, C − ∗ rρ ε ( x ) (cid:19) ⊂ B ε ( x, r ) ⊂ B X (cid:18) x, C ∗ rρ ε ( x ) (cid:19) . Proof.
Let y ∈ B X ( x, C − ∗ r/ρ ε ( x )), for a constant C ∗ ≥ γ be ageodesic in X joining x to y and let z ∈ γ . Then, since r ≤ d ε ( x ), we have by Lemma 2.3, | xz | ≤ C − ∗ d ε ( x )2 ρ ε ( x ) ≤ C − ∗ C, with C = C ( δ, K, ε, M ) ≥
1. This then implies by the Harnack inequality (2.5), ρ ε ( z ) ≍ e C − ∗ Cε ρ ε ( x ) . Choosing C ∗ large enough that e C − ∗ Cε <
2, we then obtain that ρ ε ( z ) ≍ ρ ε ( x ) , for z ∈ γ . We conclude that d ε ( x, y ) ≤ Z γ ρ ε ds ≤ ρ ε ( x ) | xy |≤ C − ∗ r< r, provided we take C ∗ >
2. This gives the inclusion on the left side of (2.8).For the inclusion on the right side of (2.8), let y ∈ B ε ( x, r ) and let γ ε be a geodesic in X ε connecting x to y . For z ∈ γ ε we then have z ∈ B ε ( x, r ) and therefore d ε ( z ) ≥ d ε ( x )by the triangle inequality since r ≤ d ε ( x ). Applying Lemma 2.3, we then have ρ ε ( z ) ≥ C − d ε ( z ) ≥ C − d ε ( x ) ≥ C − ρ ε ( x ) , OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 9 for a constant C = C ( δ, K, ε, M ) ≥
1. Using this we conclude that r > d ε ( x, y ) = Z γ ε ρ ε ds ≥ C − ρ ε ( x ) | xy | , since ℓ ( γ ε ) ≥ | xy | . Choosing C ∗ to be greater than the constant C on the right side of thisinequality, we then conclude that | xy | < C ∗ rρ ε ( x ) , which gives the right side inclusion in (2.8). (cid:3) Following [2], the balls B ε ( x, r ) for x ∈ X ε , 0 < r ≤ d ε ( x ) will often be referred to as subWhitney balls .The second claim adapts [2, Lemma 4.8] to our setting. The proof given in [2] stronglyrelies on the uniformization X ε being bounded in their setting, so when b ∈ B ( X ) we willhave to take an approach that is somewhat different. Lemma 2.5.
There is a constant κ = κ ( δ, K, ε, M ) such that for every x ∈ ¯ X ε and every < r ≤ X ε we can find a ball B ε ( z, κ r ) ⊂ B ε ( x, r ) with d ε ( z ) ≥ κ r .Proof. The claim for b ∈ D ( X ) of the form b z ( x ) = d ( x, z ) for some z ∈ X follows fromrepeating the proof of [2, Lemma 4.8] in our setting. For b ∈ D ( X ) of the form b ( x ) = d ( x, z ) + s for some z ∈ X , s ∈ R , the claim then follows by observing that X ε = e − s X ε,z ,i.e., X ε is obtained by scaling by a factor of e − s the metric on the conformal deformation of X by ρ ε,z ( x ) = e − ε | xz | . We can thus assume that b ∈ B ( X ) with basepoint ω ∈ ∂X , whichimplies that diam X ε = ∞ .Let x ∈ ¯ X ε and r > y → d ε ( y ) on X ε is continuous, positive,unbounded (since X ε is unbounded) and takes values arbitrarily close to 0 since d ε ( x n ) → { x n } in X ε converging to a point of ∂X ε . Since X ε is connectedwe can then conclude by the intermediate value theorem that d ε ( X ε ) = (0 , ∞ ), i.e., for any r > z ∈ X ε such that d ε ( z ) = r . For our given r > z and let σ be a geodesic in X joining x to z ; recall that we can consider points x ∈ ∂X ε as points of ∂ ω X through the identification ∂X ε ∼ = ∂ ω X . Then σ is an A -uniformcurve in X ε with A = A ( δ, K, ε, M ) ≥
1. Since σ does not have ω as an endpoint, it hasfinite length ℓ ε ( σ ) ≤ Ad ε ( x, z ) in X ε . We parametrize σ by d ε -arclength and orient it from x to z .We first assume that ℓ ε ( σ ) ≥ r . In this case we set z = σ ( r ). Then since σ is A -uniformwe have d ε ( z ) ≥ r A and B ε (cid:16) z, r A (cid:17) ⊂ B ε (cid:16) x, r r A (cid:17) ⊂ B ε ( x, r ) . So in this case we can use any κ ≤ A .Now consider the case in which ℓ ε ( σ ) < r . We then set z = z and observe that B ε (cid:16) z , r (cid:17) ⊂ B ε (cid:16) x, ℓ ε ( σ ) + r (cid:17) ⊂ B ε ( x, r ) . By construction we have d ε ( z ) = r . Thus in this case any κ ≤ will work. By combiningthese two cases we can then set κ = A , noting that A ≥ (cid:3) The conclusion of Lemma 2.5 is closely related to the corkscrew condition for domains inmetric spaces. See [3, Definition 2.4]. Doubling for uniformized measures
In this section we will prove Theorem 1.2 and lay some of the groundwork for proving ourother theorems. We will frequently make use of the following consequence of the doublingestimate (1.5) for a metric measure space (
X, d, µ ): if µ is doubling on balls of radius atmost R with constant C µ and 0 < r ≤ R ≤ R then(3.1) µ ( B X ( x, R )) ≍ C µ ( B X ( x, r )) , with constant C depending only on C µ and the ratio R/r . This estimate follows by iteratingthe estimate (1.5) and noting that µ ( B X ( x, R )) ≥ µ ( B X ( x, r )) since B X ( x, r ) ⊂ B X ( x, R ).We will require the following proposition from [2], which is stated there in a more generalform. Proposition 3.1. [2, Proposition 3.2]
Let ( X, d ) be a geodesic metric space and let µ be aBorel measure on X that is doubling on balls of radius at most R with doubling constant C µ . Then for any R > the measure µ is doubling on balls of radius at most R , withdoubling constant depending only on R /R and C µ . Thus if µ is doubling on balls of radius at most R then given any R > µ is also doubling on balls of radius at most R , at the cost of increasing the uniformlocal doubling constant of µ by an amount depending only on R /R and C µ .We now describe the setting of this section. We begin with a proper geodesic δ -hyperbolic X together with a function b ∈ ˆ B ( X ) with basepoint ω such that X is K -roughly starlikefrom ω . We let ε > ρ ε is a GH-density for X withconstant M . As in the previous section we write X ε for the uniformization of X , d ε for thedistance on X ε , etc. We let µ be a Borel regular measure on X such that there is an R > µ is doubling on balls of radius at most R with doubling constant C µ . For agiven β > µ β = µ β,b on ¯ X ε as in (1.6).In the claims in the rest of this section all implicit constants will depend only on δ , K , ε , M , β , R , and C µ . We will refer to this collection of seven parameters as the data . We willrefer to the specific parameters δ , K , ε , M , and β as the uniformization data and say that aconstant depends only on the uniformization data if it depends only on these five parameters.At several points we will need to increase the radius R by an amount depending only on theuniformization data in order to ensure that µ is doubling at a larger scale using Proposition3.1. When we do this we will also need to increase C µ by a corresponding amount dependingonly on the uniformization data and the local doubling constant C µ for µ . Remark . We will also often refer to just the four parameters δ , K , ε , and M as theuniformization data. It will be clear from context when β can and cannot be excluded fromthe list.The first part of this section will be devoted to proving the following technical criterionfor µ β to be doubling on ¯ X ε . Throughout this section we let κ = κ ( δ, K, ε, M ) be definedas in Lemma 2.5 and set κ = κ / Proposition 3.3.
Suppose that there is a constant C ≥ such that for any ξ ∈ ∂X ε , r > , and z ∈ X we have that whenever B ε ( z, κ r ) ⊂ B ε ( ξ, r ) and d ε ( z ) ≥ κ r , (3.2) µ β ( B ε ( ξ, r )) ≤ C r β/ε µ ( B X ( z, R )) . Then µ β is doubling on ¯ X ε with doubling constant C µ β depending only on the data and C . OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 11
We have formulated Proposition 3.3 in the manner that is most convenient for us to verifyin practice, however this comes at the cost of obscuring the connection of the inequality (3.2)to the doubling property for µ β . In order to prove Proposition 3.3 we will need a series oflemmas that establish this connection. Our first claim corresponds to [2, Lemma 4.5]. Itprovides an estimate on the measure of subWhitney balls in X ε . Lemma 3.4.
Let x ∈ X and < r ≤ d ε ( x ) . Then µ β ( B ε ( x, r )) ≍ ρ β ( x ) µ (cid:18) B X (cid:18) x, rρ ε ( x ) (cid:19)(cid:19) , with comparison constant depending only on the data.Proof. By Lemma 2.3 we have for all y ∈ B ε ( x, r ),(3.3) ρ β ( y ) = ρ ε ( y ) β/ε ≍ d ε ( y ) β/ε ≍ d ε ( x ) β/ε ≍ ρ β ( x ) , with the comparison d ε ( y ) ≍ d ε ( x ) following from the condition on r . Applying Lemma2.4 and the chain of comparisons (3.3), we conclude that µ β ( B ε ( x, r )) ≍ ρ β ( x ) µ ( B ε ( x, r )) . ρ β ( x ) µ (cid:18) B X (cid:18) x, C ∗ rρ ε ( x ) (cid:19)(cid:19) , with C ∗ = C ∗ ( δ, K, ε, M ) being the constant from Lemma 2.4. A similar argument usingthe other inclusion from Lemma 2.4 shows that µ β ( B ε ( x, r )) & ρ β ( x ) µ (cid:18) B X (cid:18) x, C − ∗ rρ ε ( x ) (cid:19)(cid:19) . We thus conclude that(3.4) ρ β ( x ) µ (cid:18) B X (cid:18) x, C − ∗ rρ ε ( x ) (cid:19)(cid:19) . µ β ( B ε ( x, r )) . ρ β ( x ) µ (cid:18) B X (cid:18) x, C ∗ rρ ε ( x ) (cid:19)(cid:19) The condition on r implies that(3.5) rρ ε ( x ) ≤ d ε ( x ) ρ ε ( x ) ≤ C, with C depending only on the uniformization data by Lemma 2.3. By Proposition 3.1 wecan, at the cost of increasing the local doubling constant C µ of µ by an amount dependingonly on the data, assume that R > CC ∗ for the constant C in inequality (3.5) and theconstant C ∗ in Lemma 2.4. Then the comparison (3.1) allows us to conclude that µ (cid:18) B X (cid:18) x, C − ∗ rρ ε ( x ) (cid:19)(cid:19) ≍ µ (cid:18) B X (cid:18) x, rρ ε ( x ) (cid:19)(cid:19) ≍ µ (cid:18) B X (cid:18) x, C ∗ rρ ε ( x ) (cid:19)(cid:19) . Combining this comparison with inequality (3.4) proves the lemma. (cid:3)
By combining Lemma 3.4 with Lemma 2.5 we obtain the following estimate for µ β ( B ε ( x, r ))when 0 < r ≤ d ε ( x ). We recall that κ = κ /
10, where κ is defined as in Lemma 2.5. Lemma 3.5.
Let x ∈ X and < r ≤ d ε ( x ) . Let z ∈ X be given such that B ε ( z, κ r ) ⊂ B ε ( x, r ) and d ε ( z ) ≥ κ r . Then µ β ( B ε ( x, r )) ≍ µ β ( B ε ( z, κ r )) , with comparison constants depending only on the data. Proof.
By Lemmas 2.3 and 2.4 we have(3.6) | xz | ≤ C ∗ rρ ε ( x ) ≤ C ∗ d ε ( x )2 ρ ε ( x ) . , with implied constant depending only on the uniformization data, where C ∗ is the constantfrom Lemma 2.4. Since z ∈ B ε ( x, r ) and r ≤ d ε ( x ), we conclude that we have d ε ( z ) ≍ d ε ( x ). We thus obtain from Lemma 2.3 that ρ ε ( z ) ≍ ρ ε ( x ) with comparison constantdepending only on the uniformization data. Since d ε ( z ) ≥ κ r , we have by Lemma 2.3that(3.7) 1 & κ rρ ε ( z ) ≍ rρ ε ( z ) ≍ rρ ε ( x ) ≍ C ∗ rρ ε ( x ) , with all implied constants depending only on the uniformization data, since κ depends onlyon the uniformization data. We can thus apply Proposition 3.1 to conclude that we canassume that µ is doubling on balls of radius at most any of the terms appearing in (3.7), atthe cost of increasing the local doubling constant of µ by an amount depending only on thedata. It follows that µ (cid:18) B X (cid:18) z, κ rρ ε ( z ) (cid:19)(cid:19) ≍ µ (cid:18) B X (cid:18) z, rρ ε ( z ) (cid:19)(cid:19) ≍ µ (cid:18) B X (cid:18) z, rρ ε ( x ) (cid:19)(cid:19) ≍ µ (cid:18) B X (cid:18) x, C ∗ rρ ε ( x ) (cid:19)(cid:19) ≍ µ (cid:18) B X (cid:18) x, rρ ε ( x ) (cid:19)(cid:19) with implied constants depending only on the data. The third comparison above followsfrom the fact that z ∈ B X (cid:16) x, C ∗ rρ ε ( x ) (cid:17) by (3.6). Since the comparison ρ β ( z ) ≍ ρ β ( x ) followsfrom the comparison ρ ε ( z ) ≍ ρ ε ( x ) (with comparison constants depending only on theuniformization data), applying Lemma 3.4 to B ε ( z, κr ) and B ε ( x, r ) (note that κ r ≤ d ε ( z )by assumption) then gives µ β ( B ε ( z, κ r )) ≍ µ β ( B ε ( x, r )) , with comparison constants depending only on the data. (cid:3) Our final lemma estimates the right side of inequality (3.2) in terms of µ β ( B ε ( z, κ r )).The reason for choosing 5 r as the upper bound for d ε ( z ) will be clear in the proof ofProposition 3.3. Lemma 3.6.
Let z ∈ X and r > be such that κ r ≤ d ε ( z ) < r . Then (3.8) µ β ( B ε ( z, κ r )) ≍ r β/ε µ ( B X ( z, R )) , with comparison constant depending only on the data.Proof. The assumptions imply that d ε ( z ) ≍ r , hence ρ β ( z ) ≍ r β/ε by Lemma 2.3, withcomparison constants depending only on the uniformization data since κ depends only onthe uniformization data. Thus by Lemma 3.4 we have µ β ( B ε ( z, κ r )) ≍ r β/ε µ (cid:18) B X (cid:18) z, κ rρ ε ( z ) (cid:19)(cid:19) , OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 13 with comparison constant depending only on the data. Since ρ ε ( z ) ≍ d ε ( z ) ≍ r , we have κ rρ ε ( z ) ≍ C ′ C ′ depending only on the uniformization data. Using Proposition3.1 we can assume that µ is doubling on balls of radius at most C ′ R , at the cost of increasingthe doubling constant by an amount depending only on the data. From this we concludethat the comparison (3.8) holds. (cid:3) We can now prove Proposition 3.3.
Proof of Proposition 3.3.
We split the proof of the doubling property for µ β into two casesdepending on the center x ∈ ¯ X ε of the ball. The first case is that in which 0 < r ≤ d ε ( x ),which implies in particular that x ∈ X ε . Then we can apply Lemma 3.4 to both B ε ( x, r )and B ε ( x, r ). We conclude that(3.9) µ β ( B ε ( x, r )) ≍ µ (cid:18) B X (cid:18) x, rρ ε ( x ) (cid:19)(cid:19) ≍ µ (cid:18) B X (cid:18) x, rρ ε ( x ) (cid:19)(cid:19) ≍ µ β ( B ε ( x, r )) , with comparison constants depending only on the data. To justify the middle comparisonin (3.9), we observe that since 2 r ≤ d ε ( x ) we have by Lemma 2.3 that each of the middletwo balls in X in (3.9) on the right side of this inequality have radius at most C ′ for someconstant C ′ depending only on the uniformization data. By Proposition 3.1 we can assumethat µ is doubling on balls of radius at most C ′ , at the cost of increasing the doublingconstant of µ by an amount depending only on the data. This gives the desired doublingestimate for the right side of (3.9). We note that this first case does not require the use ofthe assumed inequality (3.2).The second case is that in which d ε ( x ) < r . We can then find a point ξ ∈ ∂X ε suchthat B ε ( x, r ) ⊂ B ε ( ξ, r ). We then use Lemma 2.5 to choose a point z ∈ X ε such that B ε ( z, κ r ) ⊂ B ε ( x, r ) and d ε ( z ) ≥ κ r . Then we must have d ε ( z ) < r since z ∈ B ε ( ξ, r ).Since B ε ( z, κ r ) ⊂ B ε ( ξ, r ) and κ > κ , we conclude from Lemma 3.6 and the assumedinequality (3.2) that(3.10) µ β ( B ε ( x, r )) ≍ µ β ( B ε ( z, κ r )) , with comparison constant depending only on the data and C . Since we also have B ε ( x, r ) ⊂ B ε ( ξ, r ) and κ = 10 κ , the same combination of Lemma 3.6 and (3.2) also shows that(3.11) µ β ( B ε ( x, r )) ≍ µ β ( B ε ( z, κ r )) , with comparison constant depending only on the data and C . Combining (3.10) and (3.11)gives the desired doubling estimate in this second case. (cid:3) We will now prove Theorem 1.2 by showing, in analogy to [2, Proposition 4.7], that µ β is always doubling on ¯ X ε for β sufficiently large. We will need the following refinement ofProposition 3.1. Lemma 3.7. [2, Lemma 3.5]
Let ( X, d ) be a geodesic metric space and let µ be a measureon X that is doubling on balls of radius at most R with constant C µ . Let n ∈ N be a giveninteger.(1) For x, y ∈ X and < r ≤ R satisfying d ( x, y ) < nr , we have µ ( B X ( x, r )) ≤ C nµ µ ( B X ( y, r )) . (2) For < r ≤ R , every ball B ⊂ X of radius nr can be covered by at most C n +4) / µ balls of radius r . Proof of Theorem 1.2.
We will prove this theorem using the criterion of Proposition 3.3.All implied constants will depend only on the data, meaning only on the parameters δ , K , ε , M , β , R and C µ , unless otherwise noted. Let ξ ∈ ∂X ε , z ∈ X ε , and r > B ε ( z, κ r ) ⊂ B ε ( ξ, r ) and d ε ( z ) ≥ κ r , where we recall that κ = κ /
10 depends onlyon the uniformization data. We then have ρ β ( z ) ≍ r β/ε by the proof of Lemma 3.6. Wedefine for n ≥ A n = { x ∈ B ε ( ξ, r ) ∩ X ε : e − εn r ≤ d ε ( x ) < e − ε ( n − r } . Since x ∈ B ε ( ξ, r ) implies that d ε ( x ) < r , we have B ε ( ξ, r ) ∩ X ε = S ∞ n =1 A n . Since µ β isextended to ∂X ε by setting µ β ( ∂X ε ) = 0, we conclude that µ β ( B ε ( ξ, r )) = ∞ X n =1 µ β ( A n ) . For any given x ∈ A n we either have | xz | < | xz | ≥
1. In the second case we useLemma 2.2 to obtain e ε | xz | = e − ε ( x | z ) b ρ ε ( x ) ρ ε ( z ) ≍ d ε ( x, z ) d ε ( x ) d ε ( z ) ≤ ( d ε ( x, ξ ) + d ε ( ξ, z )) κ e − εn r ≤ e εn κ . e εn , with implied constant depending only on δ , K , ε , and M . We then conclude that | xz | ≤ n + c , with c = c ( δ, K, ε, M ) ≥
0. Since this inequality trivially holds with c = 0 when | xz | <
1, we in fact obtain the inequality | xz | ≤ n + c in both cases. We then choose n = n ( δ, K, ε, M ) to be the minimal integer such that n ≥ c . Then for n ≥ n wehave | xz | ≤ n for x ∈ A n . For 1 ≤ n ≤ n we then have | xz | ≤ n and therefore A n ⊂ B X ( z, n ). By Proposition 3.1 we can then assume that µ is doubling on balls ofradius at most 2 n in X , at the cost of increasing the doubling constant by an amountdepending only on δ , K , ε , and M . We conclude that µ ( B X ( z, n )) ≍ µ ( B X ( z, R )) , with comparison constant depending only on the data. By the Harnack inequality (2.5) for ρ β we conclude for x ∈ B X ( z, n ) that ρ β ( x ) ≍ ρ β ( z ) ≍ r β/ε . Putting all of this together,we conclude that µ β n [ n =1 A n ! ≤ µ β ( B X ( z, n )) . r β/ε µ ( B X ( z, R )) . We now consider the case n > n , for which we have | xz | ≤ n whenever x ∈ A n .We apply Proposition 3.1 to ensure that µ is doubling on balls of radius at most R =max { R , R + 2 } . The doubling constant C ′ µ for µ on balls of radius at most R thendepends only on R and C µ . In particular C ′ µ does not depend on β . Applying (2) of OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 15
Lemma 3.7, we cover A n ⊂ B X ( z, n ) with N n . e αn many balls B n,j of radius R , where α = α ( R , C µ ) = 76 log C ′ µ . We set β := 3 α and assume that β ≥ β . Note that β = β ( R , C µ ) depends only on R and C µ .We can clearly assume that each ball B n,j intersects A n , from which we conclude thatthe centers x n,j of the balls B n,j satisfy | x n,j z | ≤ R + 2 n < R n, since n ≥ R > R + 2. Applying (1) of Lemma 3.7 then gives that µ ( B n,j ) ≤ ( C ′ µ ) n µ ( B X ( z, R )) ≤ e αn µ ( B X ( z, R )) , For x ∈ A n we have,(3.12) ρ β ( x ) ≍ d ε ( x ) β/ε ≍ ( e − εn r ) β/ε . The Harnack inequality (2.5) implies that ρ β ( y ) ≍ ρ β ( x n,j ) for each y ∈ B n,j (since each ball B n,j has radius R ). Furthermore, since there is some point y ∈ A n such that | x n,j y | ≤ R ,it follows from the comparison (3.12) that ρ β ( x n,j ) ≍ ( e − εn r ) β/ε . Thus we conclude that µ β ( B n,j ) ≍ ρ β ( x n,j ) µ ( B n,j ) . ( e − εn r ) β/ε µ ( B n,j ) ≤ e − βn r β/ε e αn µ ( B X ( z, R )) . By our restriction β ≥ β = 3 α , we conclude that µ β ( B n,j ) . e − αn r β/ε µ ( B X ( z, R )) . It then follows from this inequality and the bound N n . e αn that µ β ∞ [ n = n +1 A n ! ≤ ∞ X n = n +1 N n X j =1 µ β ( B n,j ) . r β/ε µ ( B X ( z, R )) ∞ X n = n +1 N n e − αn . r β/ε µ ( B X ( z, R )) ∞ X n = n +1 e − αn . r β/ε µ ( B X ( z, R )) , with the final inequality following by summing the geometric series. By combining the cases1 ≤ n ≤ n and n > n we conclude that µ β ( B ε ( ξ, r )) . r β/ε µ ( B X ( z, R )) . Since µ is doubling up to the radius R with doubling constant depending only on the data,we conclude by Proposition 3.3 that µ β is doubling on ¯ X ε with constant depending only onthe data. (cid:3) We now discuss a setting in which it is possible to obtain sharper estimates for thethreshold β above which µ β is doubling. In particular this will allow us to prove thedoubling claim in Theorem 1.4. We will keep the setting of Theorem 1.2 and then assumein addition that we have a cocompact discrete isometric action of a group Γ on X . Briefly recalling the definitions, the action by Γ is isometric if each element g ∈ Γ defines anisometry of X . It is cocompact if there is a compact set E ⊂ X such that X = S g ∈ Γ g ( E ),i.e., if X is covered by the translates of a compact subset under the action of Γ. It is discrete if for each compact subset E ⊂ X the number of g ∈ Γ such that g ( E ) ∩ E = ∅ is finite.We will assume in addition that the uniformly locally doubling measure µ is Γ-invariant,meaning that µ ( g − ( E )) = µ ( E ) for each measurable subset E ⊂ X and each g ∈ Γ. Suchmeasures often arise naturally in the context of the geometry of X ; for instance if X is a treewith bounded vertex degree and edges of unit length then we can take µ to be the measureon X induced from the 1-dimensional Lebesgue measure on the edges. Another case is thesetting of Theorem 1.4 when X is the universal cover of a closed Riemannian manifold M with sectional curvatures ≤ −
1, in which case we can take µ to be the Riemannian volumeon X . We can assume by Proposition 3.1 and the cocompactness of the action of Γ that thedoubling radius R for µ is large enough that for each x ∈ X the translates of B X ( x, R )by Γ cover X .For x ∈ X and R > N Γ ( x, R ) = { g ∈ Γ : | xg ( x ) | ≤ R } , with E denoting the cardinality of a set E . We consider the critical exponent h X definedby the following limit for a fixed x ∈ X ,(3.13) h X = lim sup R →∞ log N Γ ( x, R ) R .
Standard arguments using the cocompactness of the action of Γ show that h X does notdepend on the choice of point x ∈ X . We observe that h X can equivalently be thought ofas the limit(3.14) h X = lim sup R →∞ log µ ( B X ( x, R )) R , by observing that the translates g ( B X ( x, R )) for g ∈ Γ will cover X with bounded overlapby the uniformly local doubling property of µ and the discreteness of the action of Γ; forthis we can always enlarge the doubling radius to 2 R using Proposition 3.1 to obtain thebounded overlap property. Consequently µ ( B X ( x, R )) will be comparable to N Γ ( x, R ) when R is large, which shows that the limits (3.13) and (3.14) are the same. The volume growthentropy (1.8) considered in Theorem 1.4 is a special case of the limit (3.14).It’s clear from applying (2) of Lemma 3.7 to the limit (3.14) that we have h X < ∞ . Thusfor each h > h X and x ∈ X we have a constant C h,x ≥ R > N Γ ( x, R ) ≤ C h,x e hR . The lemma below shows that we can take the constant C h,x to be independent of x . Lemma 3.8.
For each h > h X there is a constant C h such that we have for all x ∈ X and R > , (3.16) N Γ ( x, R ) ≤ C h e hR . Proof.
Fix x ∈ X and let C h,x be the constant in (3.15). Recall that R > B X ( x, R ) by Γ cover X . For each y ∈ B X ( x, R ) we have N Γ ( y, R ) ≤ N Γ ( x, R + R ) ≤ C h,x e h ( R + R ) . OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 17
This implies that for y ∈ B X ( x, R ) we can take C h = C h,x e hR . It then follows that if y ∈ g ( B X ( x, R )) for some g ∈ Γ then N Γ ( y, R ) = N Γ ( g − ( y ) , R ) ≤ C h e hR . (cid:3) Proposition 3.9.
For each β > h X the measure µ β on ¯ X ε is doubling with doubling constant C µ β depending only on the data and the constant C h in (3.16) with h = ( β + h X ) / .Proof. We follow the outline of the proof of Theorem 1.2 above for a given β > h X , butusing the estimate (3.16) in place of the use of Lemma 3.7. As in that proof, we let ξ ∈ ∂X ε , z ∈ X ε , and r > B ε ( z, κ r ) ⊂ B ε ( ξ, r ) and d ε ( z ) ≥ κ r , wherewe recall that κ = κ /
10 depends only on the uniformization data, and we will take allimplied constants to depend only on the data and the constant C h in Lemma 3.8 with h = ( β + h X ) /
2, except where otherwise noted. We define for n ≥ A n = { x ∈ B ε ( ξ, r ) ∩ X ε : e − εn r ≤ d ε ( x ) < e − ε ( n − r } . and note as before that we have µ β ( B ε ( ξ, r )) = ∞ X n =1 µ β ( A n ) . The estimates of the proof of Theorem 1.2 then show that we have a constant c = c ( δ, K, ε, M ) ≥ n ≥ x ∈ A n we have | xz | ≤ n + c .Recall that R > B := B X ( z, R ) by Γcover X . For each n ≥ { g n,j } s n j =1 ⊂ Γ be a minimal collection of group elements suchthat the balls g n,j ( B ) cover A n for 1 ≤ j ≤ s n . By minimality we can assume that each ofthese balls intersects A n . Setting c ∗ = 2 R + c , we then have g n,j ( B ) ⊂ B X ( z, n + c ∗ ) for1 ≤ j ≤ s n . In particular g n,j ( p ) ∈ B X ( z, n + c ∗ ) for each n and j . It follows from (3.16)with h = ( β + h X ) / s n ≤ C h e h ( n + c ∗ ) ≍ e hn , since h > h X , with the second comparison making the constants implicit. On the otherhand, letting x n,j ∈ g n,j ( B ) ∩ A n be a point in this intersection, we have d ε ( x n,j ) ≍ re − εn and therefore ρ ε ( x n,j ) ≍ re − εn by Lemma 2.3. Hence ρ β ( x n,j ) ≍ r β/ε e − βn . Since all ofthe balls g n,j ( B ) have radius R and since µ is Γ-invariant, the Harnack inequality (2.5)implies that µ β ( g n,j ( B )) ≍ r β/ε e − βn µ ( g n,j ( B )) = r β/ε e − βn µ ( B ) . Thus we conclude that µ β ( A n ) . s n X j =1 µ β ( g n,j ( B )) . r β/ε s n e − βn µ ( B ) . r β/ε e ( h − β ) n µ ( B ) . Since h < β , we obtain by summing the geometric series that µ β ( B ε ( ξ, r )) . ∞ X n =1 r β/ε e ( h − β ) n µ ( B ) . r β/ε µ ( B ) . Thus the hypotheses of Proposition 3.3 hold, so we can conclude the desired doubling esti-mate for µ β . (cid:3) Propsition 3.9 proves the doubling claim of Theorem 1.4, as we will see in the next section.As Remark 3.10 below indicates, this range for the measure to be doubling is generally sharp.
Remark . For this remark we assume that we are in the setting of Theorem 1.4: we let X be a complete simply connected negatively curved Riemannian manifold with sectionalcurvatures ≤ − X . We denote the Γ-invariant Riemannian volume on X by µ , fix a point z ∈ X , andconsider the measure µ β,z on X defined for each β > dµ β,z ( x ) = e − β | xz | dµ ( x ) . By the theory of Patterson-Sullivan measures (see for instance [18, Th´eor`eme 1.7]) we have µ β,z ( X ) < ∞ if and only if β > h X . Since the conformal deformation X ,z of X withconformal factor ρ ,z ( x ) = e −| xz | is bounded, this implies that µ β,z is not doubling on X ,z when β ≤ h X . If we consider the renormalizations ¯ µ β,z = µ β,z ( X ) − µ β,z of µ β,z for β > h X as a measure on ¯ X ,z ∼ = X ∪ ∂X and take the limit as β → h X then these measures convergein the weak* topology to a measure ν z on X ∪ ∂X that is supported on ∂X ; here we areusing that the induced topology on X ∪ ∂X from the identification ¯ X ,z ∼ = X ∪ ∂X coincideswith the standard cone topology on X ∪ ∂X , see [4, Remark 4.14(b)]. This measure ν z willbe uniformly comparable with the Patterson-Sullivan measure on ∂X based at z .4. Poincar´e inequalities for uniformized measures
We begin this section by formally introducing Poincar´e inequalities. We let (
X, d, µ ) bea metric measure space with the property that 0 < µ ( B ) < ∞ for all balls B ⊂ X . For ameasurable subset E ⊂ X satisfying 0 < µ ( E ) < ∞ and a function u that is µ -integrableover E we write(4.1) u E = − Z E u dµ = 1 µ ( E ) Z E u dµ for the mean value of u over E . Let u : X → R be given. A Borel function g : X → [0 , ∞ ]is an upper gradient for u if for each rectifiable curve γ joining two points x, y ∈ X we have | u ( x ) − u ( y ) | ≤ Z γ g ds. A measurable function u : X → R is integrable on balls if for each ball B ⊂ X we have that u is integrable over B . For a given p ≥ X supports a p -Poincar´e inequality ifthere are constants λ ≥ C PI > u : X → R that is integrable on balls, for each ball B ⊂ X , and each upper gradient g of u we have(4.2) − Z B | u − u B | dµ ≤ C PI diam( B ) (cid:18) − Z λB g p dµ (cid:19) /p , for a constant C PI >
0. The constant λ is called the dilation constant . If there is a constant R > R then we will say that X supports a p -Poincar´e inequality on balls of radius at most R . We will also say that X supports a uniformly local p -Poincar´e inequality . By H¨older’s inequality a metric measurespace that supports a p -Poincar´e inequality also supports a q -Poincar´e inequality for each q ≥ p , and the same is true in regards to supporting a uniformly local p -Poincar´e inequality.For this section we carry over the same standing hypotheses and notation as discussedat the start of Section 3. We will assume in addition that we are given p ≥ X is equipped with a uniformly locally doubling measure µ thatsupports a p -Poincar´e inequality on balls of radius at most R , where R is the same radiusup to which µ is doubling on X . We note that Proposition 3.1 implies that there is no lossof generality in assuming that these two radii are the same. We will also assume that µ β OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 19 is doubling on ¯ X ε for some constant C µ β . We will show under these hypotheses that themetric measure space ( ¯ X ε , d ε , µ β ) supports a p -Poincar´e inequality with dilation constant λ = 1 and constant C ∗ PI depending only on the uniformization data and the constants R , C µ , C β , p , λ , and C PI associated to the uniformly local doubling property of µ , the globaldoubling of µ β , and the uniformly local p -Poincar´e inequality on X . In particular this provesTheorem 1.3.The proof splits into two steps. In the first step we show that the p -Poincar´e inequality(4.2) holds on sufficiently small subWhitney balls in the metric measure space ( X ε , d ε , µ β ).The proof is essentially identical to [2, Lemma 6.1]. In the statement and proof of Lemma4.2 “the data” refers to the uniformization data and the constants R , C µ , p , λ , and C PI .For Lemma 4.2 we do not need to assume that µ β is doubling. We will require the followingeasy lemma. Lemma 4.1. [1, Lemma 4.17]
Let u : X → R be integrable, let p ≥ , let α ∈ R , and let E ⊂ X be a measurable set with < µ ( E ) < ∞ . Then (cid:18) − Z E | u − u E | p dµ (cid:19) /p ≤ (cid:18) − Z E | u − α | p dµ (cid:19) /p Lemma 4.2.
There exists c > depending only on the uniformization data and R suchthat for all x ∈ X ε and all < r ≤ c d ε ( x ) the p -Poincar´e inequality (4.2) for µ β holds onthe ball B ε ( x, r ) with dilation constant ˆ λ and constant ˆ C PI depending only on the data.Proof. Put B ε = B ε ( x, r ) with 0 < r ≤ c d ε ( x ), where 0 < c ≤ is a constant to bedetermined. Let C ∗ be the constant of Lemma 2.4. We choose c > c C ∗ ≤ . We conclude by applying Lemma 2.4 twice that(4.3) B ε ⊂ B := B X (cid:18) x, C ∗ rρ ε ( x ) (cid:19) ⊂ B ε (cid:0) x, C ∗ r (cid:1) = ˆ λB ε , with ˆ λ = C ∗ , since C ∗ r ≤ c C ∗ d ε ( x ) ≤ d ε ( x ) . Moreover by (3.3) we see that for all y ∈ ˆ λB ε we have ρ β ( y ) ≍ ρ β ( x ) with comparisonconstant depending only on the uniformization data.Now let u be a function on X ε that is integrable on balls and let g ε be an upper gradientof u on X ε . By the same basic calculation as in [2, (6.3)] we have that g := g ε ρ ε is an uppergradient of u on X . For c sufficiently small (depending only on the uniformization dataand R ) we will have by Lemma 2.3 that C ∗ rρ ε ( x ) ≤ C ∗ c d ε ( x ) ρ ε ( x ) ≤ R . Thus the p -Poincar´e inequality (4.2) (for µ ) holds on B . Since ρ β ( y ) ≍ ρ β ( x ) on ˆ λB ε withcomparison constant depending only on the uniformization data (by (3.3)) we have that(4.4) µ β ( B ) ≍ ρ β ( x ) µ ( B ) , with comparison constant depending only on the uniformization data, and the same com-parison holds with either B ε or ˆ λB ε replacing B . Writing u B,µ = − R B u dµ , we conclude byusing the inclusions of (4.3), the measure comparison (4.4), and the p -Poincar´e inequality for µ on B , − Z B ε | u − u B,µ | dµ β . − Z B | u − u B,µ | dµ ≤ C PI C ∗ rρ ε ( x ) (cid:18) − Z B g p dµ (cid:19) /p ≍ rρ ε ( x ) (cid:18) − Z B ( g ε ρ ε ) p dµ β (cid:19) /p . r (cid:18) − Z ˆ λB ε g pε dµ β (cid:19) /p , where all implied constants depend only on the data. By Lemma 4.1 we can replace u B,µ with u B ε ,µ β = − R B ε u dµ β on the left to conclude the proof of the lemma. (cid:3) The second part of the proof is the following key proposition.
Proposition 4.3. [2, Proposition 6.3]
Let Ω be an A -uniform metric space equipped witha doubling measure ν such that there is a constant < c < for which the p -Poincar´einequality (4.2) holds for fixed constants C PI and λ on all subWhitney balls B of the form B = B Ω ( x, r ) with x ∈ Ω and < r ≤ c d Ω ( x ) . Then the metric measure space (Ω , d, ν ) supports a p -Poincar´e inequality with dilation constant A and constant C ′ PI depending onlyon A , c , p , C PI , λ , and the doubling constant C ν for ν . This proposition is stated for bounded A -uniform metric spaces in [2] but the proof workswithout modification for unbounded A -uniform metric spaces provided that the doublingproperty of ν holds at all scales and the p -Poincar´e inequality on subWhitney balls hold atall appropriate scales.We can now verify the global p -Poincar´e inequality on ¯ X ε , which proves Theorem 1.3.Below “the data” includes all the constants from Lemma 4.2 as well as the doubling constant C µ β for µ β . Proof of Theorem 1.3.
By Lemma 4.2 there is a c > p -Poincar´e inequality holds on subWhitney balls of the form B ε ( x, r ) with 0 < r ≤ c d ε ( x ) for x ∈ X , with uniform constants ˆ C PI and ˆ λ . Since ( X ε , d ε ) is an A -uniform metricspace with A = A ( δ, K, ε, M ) and we assumed µ β is globally doubling on X ε with constant µ β , it follows from Proposition 4.3 that the metric measure space ( X ε , d ε , µ β ) supports a p -Poincar´e inequality with constant C ′ PI depending only on the data and dilation constant A . Since X ε is geodesic it follows that the p -Poincar´e inequality (4.2) in fact holds withdilation constant 1, with constant C ∗ PI depending only on the data [13, Theorem 4.18].By [14, Lemma 8.2.3] we conclude that the completion ( ¯ X ε , d ε , µ β ) (with µ β ( ∂X ε ) = 0)also supports a p -Poincar´e inequality with constants depending only on the constants forthe p -Poincar´e inequality on X ε and the doubling constant of µ β . Since ¯ X ε is also geodesicit follows by the same reasoning [13, Theorem 4.18] that we can take the dilation constantto be 1 in this case as well. (cid:3) We can now prove Theorem 1.4 as well.
Proof of Theorem 1.4.
Let X be a complete simply connected n -dimensional Riemannianmanifold X with sectional curvatures ≤ − X is δ -hyperbolic with δ = δ ( H ) being the same asthat of the hyperbolic plane H of constant negative curvature − µ be the OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 21
Γ-invariant Riemannian volume on X . The space X is 0-roughly starlike from any pointof X ∪ ∂X since any geodesic γ : I → X defined on any interval I ⊂ R can be uniquelyextended to a full geodesic line γ : R → X . By [9, Theorem 1.10] the densities ρ ,b for b ∈ ˆ B ( X ) are GH-densities with a uniform constant M . Thus we can apply the results ofthe previous sections here with this constant M and δ = δ ( H ), K = 0, and ε = 1.Choose R > x ∈ X the translates of the ball B X ( x, R )by Γ cover X . On each such ball B X ( x, R ) the Riemannian metric on X is biLipschitz tothe standard Euclidean metric on the unit ball in R n with biLipschitz constant independentof x (by the cocompactness of Γ) and the Riemannian volume is uniformly comparable tothe standard n -dimensional Lebesgue measure. Since R n equipped with the n -dimensionalLebesgue measure is a doubling metric measure space that supports a 1-Poincar´e inequality[13, Chapter 4], it follows that X equipped with µ is uniformly locally doubling and supportsa uniformly local 1-Poincar´e inequality. We remark that all of the parameters considered sofar are independent of the choice of b ∈ ˆ B ( X ).We conclude by Proposition 3.9 that for each β > h X the metric measure space ( ¯ X ,b , d ,b , µ β,b )is doubling with a uniform doubling constant C µ β independent of the choice of b ∈ ˆ B ( X ).The 1-Poincar´e inequality on ( X ,b , d ,b , µ β,b ) and ( ¯ X ,b , d ,b , µ β,b ) then follows from Theo-rem 1.3. (cid:3) Uniform inversion
In this section we consider a procedure that we will call uniform inversion that can beused to convert bounded uniform metric spaces into unbounded uniform metric spaces andvice versa. This procedure can be thought of as a variation of the inversion procedureconsidered in [15] that is specialized to the context of uniform metric spaces. We show thatthis procedure can be extended to measures in such a way that it preserves the doublingproperty and p -Poincar´e inequalities for a given p ≥
1. For general metric measure spacesit was shown by Li and Shanmugalingam [16] that the doubling property can be preservedunder sphericalization and inversion, however they were only able to obtain preservationof p -Poincar´e inequalities under the additional assumption that the space was annularlyquasiconvex. This condition excludes many uniform metric spaces such as those that areobtained by uniformizing trees. With a weaker assumption Durand-Cartagena and Li [11]showed that p -Poincar´e inequalities can be preserved once p is sufficiently large. Using theresults of the previous sections we will show that uniform inversion preserves p -Poincar´einequalities for all p ≥ , d ) be an A -uniform metric space, A ≥
1. We denote the distance to the metricboundary of Ω by d ( x ) := d Ω ( x ) for x ∈ Ω. The quasihyperbolic metric on Ω is defined by,for x, y ∈ Ω,(5.1) k ( x, y ) = inf Z γ dsd ( γ ( s )) , where the infimum is taken over all rectifiable curves joining x to y . The metric space Y = (Ω , k ) is called the quasihyperbolization of the metric space (Ω , d ). We note that Y can equivalently be thought of as the conformal deformation of Ω with conformal factor ρ ( x ) = d ( x ) − . The quasihyperbolication Y is a proper geodesic δ -hyperbolic space by [4,Theorem 3.6] with δ = δ ( A ) depending only on A .To precisely state our claims below we introduce the following ratio when Ω is bounded,(5.2) φ (Ω) := diam Ωdiam ∂ Ω , where we define φ (Ω) = ∞ if ∂ Ω contains only one point. Until the end of this sectionwe will always assume that φ (Ω) < ∞ if Ω is bounded, i.e., that ∂ Ω contains at least twopoints.
Remark . The case of bounded Ω with ∂ Ω containing only one point is rather degenerateso we will not discuss it here. For instance if Ω = [0 ,
1) then its quasihyperbolization Y isisometric to [0 , ∞ ), and a Busemann function b on [0 , ∞ ) based at the only point ∞ in theGromov boundary of [0 , ∞ ) is given by b ( t ) = − t for t ∈ [0 , ∞ ). By direct calculation wethen see for every ε > Y ε,b is also isometric to [0 , ∞ ). In particular Y ε,b is actually acomplete metric space, so it can’t be a uniform metric space.When Ω is unbounded there is a constant K = K ( A ) depending only on A such that Y is K -roughly starlike from any point of Y ∪ ∂Y , while when Ω is bounded there is a constant K = K ( A ) such that Y is K -roughly starlike from any z ∈ Ω such that d ( z ) = sup x ∈ Ω d ( x ),and a constant K ′ = K ′ ( A, φ (Ω)) depending only on A and the ratio φ (Ω) such that Y is K ′ -roughly starlike from any point of Y ∪ ∂Y [8, Proposition 3.3]. The dependence of K ′ on φ (Ω) in the bounded case is necessary by [8, Example 3.4].From the discussion after inequality (1.7) we can find an ε = ε ( A ) > A (since Y is δ -hyperbolic with δ = δ ( A )) such that for any b ∈ ˆ B ( Y ) the density ρ ε,b ( x ) = e − εb ( x ) on Y is a GH-density with constant M = 20. We will fix such an ε foreach value of A for the rest of this section. Definition 5.2.
For a given b ∈ ˆ B ( Y ) we let Ω b = Y ε,b denote the conformal deformation of Y with conformal factor ρ ε,b . We will refer to the metric space Ω b as the uniform inversion of Ω based at b .The next proposition shows that uniform inversions of Ω have the properties suggestedby their name. Proposition 5.3.
Let Ω be an A -uniform metric space. Let Y = (Ω , k ) be the quasihyper-bolization of Ω . Then Ω b is an A ′ -uniform metric space for each b ∈ ˆ B ( Y ) with A ′ = A ′ ( A ) if Ω is unbounded and A ′ = A ′ ( A, φ (Ω)) if A ′ is unbounded. Furthermore Ω b is bounded ifand only if b ∈ D ( Y ) . All of the claims of Proposition 5.3 follow from applying [9, Theorem 1.1] to Y , since Y is δ = δ ( A )-hyperbolic, K = K ( A )-roughly starlike from any point of Y ∪ ∂Y if Ω isunbounded (with K = K ( A, φ (Ω)) instead if Ω is bounded) and ρ ε,b is a GH-density withconstant M = 20 (and ε = ε ( A )). In particular uniform inversion can be used to produce anunbounded uniform metric space Ω b from a bounded uniform metric space Ω by choosing b ∈ B ( Y ), and similarly can be used to produce a bounded uniform metric space Ω b froman unbounded uniform metric space Ω by choosing b ∈ D ( Y ).The primary reason to consider uniform sphericalization and inversion is that these op-erations can be extended to measures in such a way as to preserve the doubling propertyand the p -Poincar´e inequality for all p ≥
1. This comes at a price of increased complexityof these operations as opposed to the standard inversion operation, with the loss of severalnice features of the latter that are obtained in [6]. We remark that one can show thatthe identity map Ω → Ω b is always quasim¨obius for any b ∈ ˆ B ( Y ), as is true of ordinaryinversion; see [8, Proposition 4.4].Now suppose in addition that Ω is equipped with a Borel measure ν that is doubling andsatisfies 0 < ν ( B ) < ∞ for all balls B ⊂ Ω. We write C ν for the doubling constant of ν . OUBLING AND POINCAR´E INEQUALITIES FOR UNIFORMIZED MEASURES 23
For each α > µ α on Ω by dµ α ( x ) = d ( x ) − α dν ( x ) , and consider µ as a measure on Y . Then [2, Proposition 7.3] shows for each α > µ is doubling on balls of radius at most R = 1 with local doubling constant C µ α dependingonly on A and α . We let β > Y equipped with the measure µ α in relation to its uniformizationΩ b = Y ε,b with b ∈ ˆ B ( Y ) and ε = ε ( A ). We then choose β ≥ β and set ν α,β,b = ( µ α ) β,b tobe the measure obtained from µ α by applying the formula (1.6) with our chosen b ∈ ˆ B ( Y ).We consider ν α,β,b as defining a two parameter family of measures on Ω b and write d b for themetric on Ω b . Applying Theorems 1.2 and 1.3 to this family yields the following theorem. Theorem 5.4.
Let (Ω , d, ν ) be a doubling metric measure space with doubling constant C ν such that (Ω , d ) is an A -uniform metric space with φ (Ω) < ∞ if Ω is bounded. Let Y = (Ω , k ) be the quasihyperbolization of Ω . Then for each α > there is a constant β = β ( α, A, C ν ) (if Ω is unbounded) or β = β ( α, A, C ν , φ (Ω)) (if Ω is bounded) suchthat for any b ∈ ˆ B ( Y ) and any β ≥ β we have that ν α,β,b is doubling on Ω b with doublingconstant C ν α,β,b depending only on α , β , A , and C ν (and φ (Ω) if Ω is bounded).If furthermore the metric measure space (Ω , d, ν ) supports a p -Poincar´e inequality for agiven p ≥ then the metric measure space (Ω b , d b , ν α,β,b ) supports a p -Poincar´e inequalitywith constants depending only on α , β , A , C ν , p , and the constants in (4.2) (and φ (Ω) if Ω is bounded). The final claim follows from the fact that under the hypotheses of the proposition themetric measure space (
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