Extension and trace theorems for noncompact doubling spaces
aa r X i v : . [ m a t h . M G ] O c t EXTENSION AND TRACE THEOREMS FOR NONCOMPACTDOUBLING SPACES
CLARK BUTLER
Abstract.
We generalize the extension and trace results of Bj¨orn-Bj¨orn-Shanmugalingam[3] to the setting of complete noncompact doubling metric measure spaces and their uni-formized hyperbolic fillings. This is done through a uniformization procedure introducedby the author that uniformizes a Gromov hyperbolic space using a Busemann functioninstead of the distance functions considered in the work of Bonk-Heinonen-Koskela [7].We deduce several new corollaries for the Besov spaces that arise as trace spaces in thisfashion, including density of Lipschitz functions with compact support, embeddings intoH¨older spaces for appropriate exponents, and the existence of Lebesgue points quasiev-erywhere with respect to the Besov capacity under an additional reverse doubling hy-pothesis on the measure. Under this same reverse doubling hypothesis we also obtaina hyperbolic Poincar´e inequality for Besov functions that controls the mean oscillationof a Besov function over a ball through an integral over an extended ball of an uppergradient for an extension of this function to a uniformized hyperbolic filling of the space. Introduction
In recent breakthrough work of Bj¨orn-Bj¨orn-Shanmugalingam [3] an extensive array ofproperties of Besov spaces on a compact doubling metric measure space Z were establishedby exhibiting these spaces as the trace space of a Newton-Sobolev space on an associatedincomplete metric graph X ρ having Z as its boundary. These properties include densityof Lipschitz functions, the existence of quasicontinuous representatives of Besov functions,embeddings into H¨older spaces under appropriate hypotheses on the regularity exponents,and the existence of Lebesgue points quasieverywhere under an additional reverse-doublinghypothesis on the measure [3, Section 13]. In this work we extend these results to the settingof complete doubling metric measure spaces by removing the compactness hypothesis on Z .This is done through the use of a new uniformization construction for Gromov hyperbolicspaces introduced by the author in previous work [14] to similarly exhibit Z as the boundaryof an incomplete metric graph X ρ , even in the case that Z is not compact. For the compactcase this uniformization is done in [3] using the procedure of Bonk-Heinonen-Koskela [7];our construction generalizes this procedure.To state our main theorems we need to introduce some important additional concepts,which are similar in nature to those considered in [3]. For precise definitions of the notionsthat follow we refer to Sections 2, 3, and 6. A metric measure space ( Z, d, ν ) is a metricspace (
Z, d ) equipped with a Borel regular measure ν such that for every ball B ⊂ Z wehave 0 < ν ( B ) < ∞ . This space is doubling if there is a constant C ν ≥ B we have ν (2 B ) ≤ C ν ν ( B ), where 2 B denotes the ball with the same centeras B and twice the radius. Given a complete doubling metric measure space ( Z, d, ν ), inSection 6 we construct a metric graph X that is a proper geodesic Gromov hyperbolic spacesuch that Z can be canonically identified with the complement of a distinguished point ω inthe Gromov boundary ∂X of X . Such a graph is known as a hyperbolic filling of Z ; these constructions were first considered by Bonk-Kleiner [9] and Bourdon-Pajot [12] for compact Z (see also Bonk-Schramm [8] for an alternative construction known as the hyperbolic cone ).Our construction for noncompact Z is inspired by a construction due to Buyalo-Schroeder[15, Chapter 6].We then uniformize X to produce a uniform metric space X ρ such that the metric bound-ary ∂X ρ of X ρ has a canonical biLipschitz identification with Z . After a biLipschitz changeof coordinates on Z we can then assume that Z is isometrically identified with ∂X ρ . In Sec-tion 7 we lift ν to a uniformly locally doubling measure µ on the hyperbolic graph X . Weuniformize this measure for each parameter β > µ β on X ρ suchthat the resulting metric measure spaces ( X ρ , d ρ , µ β ) and ( ¯ X ρ , d ρ , µ β ) are each doublingand satisfy a 1-Poincar´e inequality, where µ β is extended to ¯ X ρ by setting µ β ( ∂X ρ ) = 0.Newtonian functions on these metric measure spaces will serve as counterparts to Besovfunctions on ( Z, d, ν ). In the process of establishing these properties of the measure µ β ,we generalize several results of Bj¨orn-Bj¨orn-Shanmugalingam [2] concerning the transfor-mation of doubling measures and Poincar´e inequalities under the Bonk-Heinonen-Koskelauniformization to the setting of our uniformization.For a given p ≥ θ >
0, and u ∈ L p loc ( Z ), the Besov norm on Z is defined by(1.1) k u k pB θp ( Z ) = Z Z Z Z | u ( x ) − u ( y ) | p d ( x, y ) pθ dν ( x ) dν ( y ) ν ( B ( x, d ( x, y ))) . We write ˜ B θp ( Z ) ⊂ L p loc ( Z ) for the subspace of all functions u such that k u k B θp ( Z ) < ∞ . Wenote that, properly speaking, (1.1) is actually a seminorm on L p loc ( Z ) since any constantfunction has Besov norm 0. We define ˇ B θp ( Z ) = L p ( Z ) ∩ ˜ B θp ( Z ) to be the subspace of ˜ B θp ( Z )consisting of those functions that are p -integrable over Z . We equip this subspace with thenorm(1.2) k u k ˇ B θp ( Z ) = k u k L p ( Z ) + k u k B θp ( Z ) , With this norm ˇ B θp ( Z ) is a Banach space by the arguments in [3, Remark 9.8]. We alwayshave ˇ B θp ( Z ) = ˜ B θp ( Z ) whenever Z is bounded. For bounded Z these Besov spaces wereintroduced by Bourdon-Pajot in a more restrictive setting in [12]. In [17] it was shown byGogatishvili-Koskela-Shanmugalingam that this definition coincides with a more classicalformulation for Besov spaces on metric spaces. We will use the notation of Bourdon-Pajotin this paper.On X ρ and its completion ¯ X ρ = X ρ ∪ Z , we will consider the Newtonian spaces definedthrough upper gradients. These spaces are based on a notion of gradient in the metricspace setting introduced by Heinonen-Koskela [20], and their formal study was initiatedby Shanmugalingam [28]. We give an abbreviated description of these spaces here; a moredetailed description can be found in Section 8. Beginning with a metric measure space(
Y, d, µ ), a Borel function g : Y → [0 , ∞ ] is an upper gradient of a function u : Y → [ −∞ , ∞ ]if for each nonconstant compact rectifiable curve γ in Y joining two points x, y ∈ Y we havethat(1.3) | u ( x ) − u ( y ) | ≤ Z γ g ds. By convention we require that R γ g ds = ∞ if u ( x ) = ±∞ or u ( y ) = ±∞ . The Newtoniannorm of u for a given p ≥ k u k N ,p ( Y ) = k u k L p ( Y ) + inf g k g k L p ( Y ) , XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 3 where the infimum is taken over all upper gradients g of u . We write ˜ N ,p ( Y ) for thecollection of functions such that k u k N ,p ( Y ) < ∞ . Similarly the Dirichlet norm of u for agiven p ≥ p -energy,(1.5) k u k D ,p ( Y ) = inf g k g k L p ( Y ) , with the infimum taken over all upper gradients g of u . We then write ˜ D ,p ( Y ) for thecollection of functions such that k u k D ,p ( Y ) < ∞ , which can equivalently be thought ofas the collection of functions u with a p -integrable upper gradient. We again note that(1.4) and (1.5) actually define seminorms on ˜ N ,p ( Y ) and ˜ D ,p ( Y ) and one must pass to aquotient to obtain a true norm. This issue will not be relevant to any of the statements wemake in this first section.We can now state our main theorem. Given p ≥ < θ <
1, the uniformizedhyperbolic filling ¯ X ρ is considered to be equipped with the uniformized measure µ β for thespecific parameter value β = p (1 − θ ). We refer to Section 7 for a precise description of themeasure µ β for each β >
0. We warn the reader that our definition of β is slightly differentthan the one used in [3]: our β corresponds to β/ǫ in their work.The linear operators below are maps between seminormed spaces, and should be under-stood as being bounded with respect to these seminorms; to simplify terminology we willgenerally refer to the seminorms (1.1), (1.4), and (1.5) as norms on ˜ B θp ( Z ), ˜ N ,p ( Y ), and˜ D ,p ( Y ) respectively, even though one must pass to a quotient of these spaces to actuallyobtain a norm. The expression (1.2) does define a genuine norm on ˇ B θp ( Z ), however. Thesematters are treated in greater detail in Section 8. Theorem 1.1.
There are bounded linear operators T : ˜ D ,p ( X ρ ) → ˜ B θp ( Z ) and P : ˜ B θp ( Z ) → ˜ D ,p ( ¯ X ρ ) such that for each f ∈ ˜ B θp ( Z ) we have T ( P f ) = f ν -a.e., i.e., T ◦ P is the identityon ˜ B θp ( Z ) .Furthermore T restricts to a bounded linear operator T : ˜ N ,p ( X ρ ) → ˇ B θp ( Z ) and thereis a truncation P of P that defines a bounded linear operator P : ˇ B θp ( Z ) → ˜ N ,p ( ¯ X ρ ) with T ◦ P being the identity on ˇ B θp ( Z ) . The operators T and P are known as trace and extension operators respectively. Werefer to Sections 9 and 10 for a description of these operators, as well as a description of theparameters on which the norms of these linear operators depend. The truncation P of P is defined in Section 10; we remark that P itself does not define a bounded linear operatorfrom ˇ B θp ( Z ) to ˜ N ,p ( ¯ X ρ ), as an adaptation of the arguments of Proposition 7.7 shows. Weemphasize that the domain of T consists of functions defined on X ρ , not on ¯ X ρ ; even though ∂X ρ has measure zero with respect to µ β , it is not at all obvious that functions in ˜ D ,p ( X ρ )and ˜ N ,p ( X ρ ) can naturally be extended to functions in ˜ D ,p ( ¯ X ρ ) and ˜ N ,p ( ¯ X ρ ). For thiswe rely on work of J. Bj¨orn and Shanmugalingam [6]. Theorem 1.1 can be interpreted assaying that the trace space of ˜ D ,p ( X ρ ) on Z is ˜ B θp ( Z ), and the trace space of ˜ N ,p ( X ρ ) on Z is ˇ B θp ( Z ) = L p ( Z ) ∩ ˜ B θp ( Z ).Theorem 1.1 yields numerous corollaries for the Besov space ˜ B θp ( Z ), all of which followfrom this theorem in essentially the same way that the corollaries of [3, Corollary 1.2] followfrom [3, Theorem 1.1]. We split these consequences into three corollaries. We refer toSection 11 for more precise formulations of these corollaries; here we have endeavored toavoid overburdening the corollaries with additional definitions. For an extensive discussionof predecessors to these corollaries we refer to the discussion after [3, Corollary 1.2]. We CLARK BUTLER remark that all of these corollaries were established in the case of compact Z in [3], howeverthe generalization from compact Z to noncompact Z is highly nontrivial due to the nonlocalnature of the Besov norm.Our first corollary shows that Lipschitz functions with compact support are dense inˇ B θp ( Z ) and that every function in ˜ B θp ( Z ) is quasicontinuous with respect to the Besov ca-pacity. Quasicontinuity is a strong refinement of Lusin’s theorem, as the Besov capacity isalways absolutely continuous with respect to the measure ν on Z (see Propositions 9.9 and10.7). For each of the corollaries below we always assume that p ≥ < θ < Corollary 1.2.
Lipschitz functions with compact support are dense in ˇ B θp ( Z ) . Each functionin ˜ B θp ( Z ) has a representative which is quasicontinuous with respect to the Besov capacity. Our second corollary concerns embeddings of Besov spaces into H¨older spaces. For thiswe will use the notion of relative lower volume decay of order Q for a doubling measure ν ,defined by the inequality (7.17) for a given exponent Q >
0. The exponent Q is also knownas a doubling dimension for ν ; we remark that the relative volume decay estimate alwaysholds with Q = log C ν , where C ν is the constant in the doubling inequality for ν , but it canalso hold for smaller Q . The notation Q β below is motivated by the equality β = p (1 − θ )that we assumed in Theorem 1.1. See also Lemma 7.6 and Proposition 11.4. Corollary 1.3.
Suppose that ν has relative lower volume decay of order Q > . Set Q β =max { , Q + p (1 − θ ) } and assume that p > Q β . Then every function in ˜ B θp ( Z ) has arepresentative that is locally (1 − Q β /p ) -H¨older continuous. Finally we consider the case in which ν additionally satisfies a reverse-doubling property(11.1), which holds in particular when Z is uniformly perfect [25, Lemma 4.1]. Corollary 1.4.
Suppose that ν has relative lower volume decay of order Q > and that ν satisfies the reverse-doubling property (11.1) . We assume further that pθ < Q and set Q ∗ = Qp/ ( Q − pθ ) . Then functions in ˜ B θp ( Z ) belong to L Q ∗ loc ( Z ) and have representativeswhich have L Q ∗ ( Z ) -Lebesgue points outside a set of zero Besov capacity. It is an interesting question what further hypotheses on Z and ν are needed to obtain f ∈ L Q ∗ ( Z ) when f ∈ ˇ B θp ( Z ). Such an estimate could be regarded as a type of Sobolevembedding theorem for ˇ B θp ( Z ). We remark that the proof of Corollary 1.4 leads to a notableinequality that we call a hyperbolic Poincar´e inequality for functions in ˜ B θp ( Z ), which westate in Proposition 11.7 at the end of the paper. Roughly speaking, this inequality showsthat upper gradients of extensions of a function f ∈ ˜ B θp ( Z ) to a function u ∈ ˜ D ,p ( ¯ X ρ )can be used to control the deviations of f from its mean on a given ball in Z in a formallysimilar manner to the classical Poincar´e inequalities that we discuss in Section 8.As we mentioned previously, this work is a direct descendant of a number of recent worksconnecting function spaces on doubling metric spaces with function spaces on hyperbolicfillings associated to the space. In addition to all of the works we mentioned previously, othernotable works include Bonk-Saksman [10] (from whom we’ve taken a significant amount ofdirect inspiration), Bonk-Saksman-Soto [11] (who consider a wider class of function spaceson Z including the Triebel-Lizorkin spaces), and Saksman-Soto [26] (who consider tracesonto Ahlfors-regular subsets of Z ). We also highlight the work of Mal´y [24], who obtainsmany similar results via different methods in the setting of John domains with compactclosure (as well as several more general settings) in metric measure spaces.The reader will find that the structure of our paper is broadly similar to that of [2] and[3]. This is because the bulk of the theory in [3] relies only on the fact that the uniformized XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 5 hyperbolic filling X ρ equipped with the lifted measure µ β for a given β > Z . Nevertheless the boundedness of Z (and consequentlyof X ρ in their setting) enters into the proofs at several critical junctures, requiring us tomodify their arguments at many steps. In particular there is no distinction between thespaces ˜ D ,p ( X ρ ) and ˜ N ,p ( X ρ ) when X ρ is bounded, as is noted in Section 8. We are alsoable to give independent (and in some cases, simpler) proofs of several of their claims dueto the more restrictive nature of our hyperbolic filling construction; our hyperbolic fillingsin general contain more edges than the hyperbolic fillings considered in [3]. A drawback ofour approach is that while we are able to recover their results on Besov spaces in [3, Section13] via specialization of our corollaries to the case of compact Z , we are not able to recovertheir results on trace and extension theorems for trees (see [3, Section 7]) due to the factthat our hyperbolic filling construction never yields a tree unless Z is a single point (see[3, Remark 7.2]; the same reasoning applies in our setting due to our parameter restriction(6.1)).We now give an overview of the structure of the paper. In Sections 2 and 3 we reviewresults from our previous work [14] as well as basic definitions concerning Gromov hyperbolicspaces and uniform metric spaces. Sections 4 and 5 are devoted to generalizing the resultsof [2] concerning global doubling and global Poincar´e inequalities to the uniformizationprocedure we introduced in [14]. In Section 2 we construct a hyperbolic filling X of acomplete doubling metric space Z and then construct a uniformization X ρ of X whoseboundary can be biLipschitz identified with Z . This primarly relies on our previous resultsin [14]. In Section 7 we construct the lifts µ β of the measure ν on Z to X ρ for each β > X ρ , d ρ , µ β ). Section 8consists of a detailed overview of the theory of Newtonian spaces for doubling metric spacessatisfying a Poincar´e inequality with an emphasis on the Newtonian capacity. We then proveTheorem 1.1 in Sections 9 and 10. Finally in Section 11 we prove Corollaries 1.2, 1.3, and1.4.We offer extensive thanks to Nageswari Shanmugalingam, who provided us with severalearly drafts of the work [3] that inspired both our work here and our previous papers [14],[13]. 2. Hyperbolic metric spaces
In this section we review some standard results regarding Gromov hyperbolic spaces, aswell as some facts regarding Busemann functions established in our previous papers [14],[13]. Standard references for this material are [15], [16].2.1.
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 if CLARK BUTLER and 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 wewrite diam( E ) = sup { d ( x, y ) : x, y ∈ E } for the diameter of 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∆ = 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 . We remark that geodesic triangles with vertices in X ∪ ∂X are 10 δ -thin by [14, Lemma 2.2].As in our previous work [14], 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 a geodesic ray γ : [0 , ∞ ) → X , the Busemann function b γ : X → R associated to γ isdefined by the limit(2.1) b γ ( x ) = lim t →∞ | γ ( t ) x | − t. XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 7
We then define(2.2) 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 [14, (1.4-1.5)] forfurther details on these definitions. The Busemann functions b ∈ B ( X ) are all 1-Lipschitzfunctions on X . For a Busemann function b of the form b = b γ + s for some s ∈ R , we definethe endpoint ω ∈ ∂X of γ to be the basepoint of b and say that b is based at ω .We next state a useful fact regarding Busemann functions. Let X be a proper geodesic δ -hyperbolic space and let b : X → R be a Busemann function based at some point ω ∈ ∂X .By [14, Lemma 2.5], if b ′ is any other Busemann function based at ω then there is a constant s ∈ R such that(2.3) b . = δ b ′ + s, with s = 0 if the geodesic rays associated to b and b ′ have the same starting point. Thusall Busemann functions based at ω differ from each other by an additive constant, up to anadditive error of 72 δ .2.2. Gromov products.
For x, y, z ∈ X the Gromov product of x and y based at z isdefined by(2.4) ( x | y ) z = 12 ( | xz | + | yz | − | xy | ) . We can also take the basepoint of the Gromov product to be any Busemann function b ∈B ( X ). For b ∈ B ( X ) the Gromov product based at b is defined by(2.5) ( x | y ) b = 12 ( b ( x ) + b ( y ) − | xy | ) . The following statements briefly summarize a more extensive discussion of Gromov prod-ucts in [14, Section 2]. In particular we give details there for how the precise forms of thesestatements follow from the corresponding statements in the literature. For these statementsit is useful to conceive of the Gromov boundary in an alternative way using Gromov prod-ucts. Fix z ∈ X . A sequence { x n } ⊂ X converges to infinity if ( x m | x n ) z → ∞ as m, n → ∞ .Two sequences { x n } and { y n } are equivalent if ( x n | y n ) z → ∞ . These notions do not de-pend on the choice of basepoint z , as can easily be checked by the triangle inequality. For aproper geodesic δ -hyperbolic space X the set of equivalence classes of sequences convergingto infinity gives an equivalent definition of the Gromov boundary ∂X , with the equivalencebeing given by sending a geodesic ray γ : [0 , ∞ ) → X to the sequence { γ ( n ) } ∞ n =0 . For ξ ∈ ∂X and a sequence { x n } that converges to infinity we will { x n } ∈ ξ if { x n } belongs tothe equivalence class of ξ . For a geodesic ray γ : [0 , ∞ ) → X in X we similarly write γ ∈ ξ if { γ ( n ) } ∞ n =0 ∈ ξ . We will also consider geodesic rays γ : ( −∞ , → X with the oppositeorientation, for which we write γ ∈ ξ if { γ ( − n ) } ∞ n =0 ∈ ξ .These notions may be extended to Busemann functions b ∈ B ( X ) based at a given point ω ∈ ∂X [15, Chapter 3]. As above a sequence { x n } converges to infinity with respect to ω if ( x m | x n ) b → ∞ as m, n → ∞ , and two sequences { x n } and { y n } are equivalent withrespect to ω if ( x n | y n ) b → ∞ as n → ∞ . These definitions do not depend on the choice ofBusemann function based at ω by (2.3). The Gromov boundary relative to ω is defined to bethe set ∂ ω X of all equivalence classes of sequences converging to infinity with respect to ω .By [15, Proposition 3.4.1] we have a canonical identification of ∂ ω X with the complement ∂X \{ ω } of ω in the Gromov boundary ∂X . We will thus use the notation ∂ ω X = ∂X \{ ω } throughout the rest of the paper. CLARK BUTLER
Gromov products based at Busemann functions b ∈ B ( X ) can be extended to points of ∂X by defining the Gromov product of equivalence classes ξ , ζ ∈ ∂X based at b to be(2.6) ( ξ | ζ ) b = inf lim inf n →∞ ( x n | y n ) b , with the infimum taken over all sequences { x n } ∈ ξ , { y n } ∈ ζ ; if b ∈ B ( X ) has basepoint ω then we leave this expression undefined when ξ = ζ = ω . As a consequence of [15, Lemma2.2.2], [15, Lemma 3.2.4], and the discussion in [14, Section 2.2], for any choices of sequences { x n } ∈ ξ and { y n } ∈ ζ we have(2.7) ( ξ | ζ ) b ≤ lim inf n →∞ ( x n | y n ) b ≤ lim sup n →∞ ( x n | y n ) b ≤ ( ξ | ζ ) b + c ( δ ) , with the constant c ( δ ) depending only on δ . One may take c ( δ ) = 600 δ . For x ∈ X and ξ ∈ ∂X the Gromov product based at b is defined analogously as(2.8) ( x | ξ ) b = inf lim inf n →∞ ( x | x n ) b , and the analogous inequality (2.7) holds with the same constants. By [14, (2.10)], for all x, y ∈ X ∪ ∂X with ( x, y ) = ( ω, ω ) we have(2.9) ( x | y ) b ≤ min { b ( x ) , b ( y ) } + c ( δ ) , where we set b ( ω ) = −∞ and b ( ξ ) = ∞ for ξ ∈ ∂ ω X . We may also take c ( δ ) = 600 δ here.We will require the following observation regarding Busemann functions in the proof ofLemma 3.7 in the next section. Lemma 2.1.
Let X be a proper geodesic Gromov hyperbolic space. Suppose that ∂X con-tains at least two points. Let b : X → R be a Busemann function based at ω ∈ ∂X . Then b ( X ) = R , i.e., b is surjective.Proof. Since ∂X contains at least two points we can find a geodesic line γ : R → X startingfrom ω and ending at some ζ ∈ ∂X with ζ = ω . By [14, Lemma 2.6] we can parametrize γ such that b ( γ ( t )) . = δ t for t ∈ R . For t ∈ R we then set ˇ b ( t ) = b ( γ ( t )). The functionˇ b : R → R is continuous and satisfies ˇ b ( t ) → ±∞ as t → ±∞ . By the intermediate valuetheorem we conclude that ˇ b ( R ) = R . By the construction of ˇ b this immediately implies that b ( X ) = R . (cid:3) The requirement that ∂X contain at least two points is necessary for Lemma 2.1 to hold,as the example X = [0 , ∞ ) shows.2.3. Visual metrics.
Let X be a proper geodesic δ -hyperbolic space. Gromov productsbased at Busemann functions b ∈ B ( X ) can be used to define visual metrics on the Gromovboundary ∂X . We refer to [15, Chapters 2-3] as well as [14, Section 2.3] for precise detailson this topic. We will summarize the results we need here. For b ∈ B ( X ) we let ω denotethe basepoint of b . We recall that we write ∂ ω X = ∂X \{ ω } .For b ∈ B ( X ) and q > ξ , ζ ∈ ∂ ω X ,(2.10) α b,q ( ξ, ζ ) = e − q ( ξ | ζ ) b . This may not define a metric on ∂ ω X , since the triangle inequality may not hold. Howeverthere is always q = q ( δ ) > δ such that for 0 < q ≤ q the function α b,q is 4-biLipschitz to a metric α on ∂ ω X . We refer to any metric α on ∂ ω X that is biLipschitzto α b,q as a visual metric on ∂ ω X based at b and refer to q as the parameter of α . We give ∂ ω X the topology associated to a visual metric based at b for any Busemann function b based at ω . When equipped with a visual metric ∂ ω X is a locally compact metric space. XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 9 Uniformization
In this section we will review the results of [14] concerning uniformizing Gromov hyper-bolic spaces with Busemann functions. We begin with two essential definitions. For thefirst one we 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 ; if ℓ ( γ ) < ∞ then we say that γ is rectifiable . For an interval I ⊂ R and t ∈ I we write I ≤ t = { s ∈ I : s ≤ t } and I ≥ t = { s ∈ I : s ≥ t } . Definition 3.1.
Let (Ω , d ) be an incomplete, locally compact metric space. For a constant A ≥ I ⊂ R , a curve γ : I → Ω with endpoints x, y ∈ Ω is A -uniform if(3.1) ℓ ( γ ) ≤ Ad ( x, y ) , and if for every t ∈ I we have(3.2) min { ℓ ( γ | I ≤ t ) , ℓ ( γ | I ≥ t ) } ≤ Ad Ω ( γ ( t )) . The metric space Ω is A -uniform if any two points in Ω can be joined by an A -uniformcurve.For the second key definition we consider a metric space ( X, d ) and a continuous function ρ : X → (0 , ∞ ). Such a positive continuous function ρ will be referred to below as a density on X . For a rectifiable curve γ in X we write ℓ ρ ( γ ) = Z γ ρ ds, for the line integral of ρ along γ . For reference below we say that X is rectifiably connected if any two points in X can be joined by a rectifiable curve. Definition 3.2.
Let (
X, d ) be a rectifiably connected metric space and let ρ : X → (0 , ∞ )be a density on X . The conformal deformation of X with conformal factor ρ is the metricspace X ρ = ( X, d ρ ) with metric(3.3) d ρ ( x, y ) = inf ℓ ρ ( γ ) , with the infimum taken over all rectifiable curves γ joining x to y .If X is geodesic then we say further that the density ρ is admissible for X with constant M ≥ x, y ∈ X and any geodesic γ joining x to y we have(3.4) ℓ ρ ( γ ) ≤ M d ρ ( x, y ) . Now consider a proper geodesic δ -hyperbolic space X . We define X to be K -roughlystarlike from a point ω ∈ ∂X if for each x ∈ X there is a geodesic line γ : R → X with γ | ( −∞ , ∈ ω and dist( x, γ ) ≤ K . We remark for use later that the rough starlikenesscondition from ω immediately implies that ∂X contains at least two points.We fix a Busemann function b based at ω and let ε > ρ ε ( x ) = e − εb ( x ) is admissible on X with constant M . Since b is 1-Lipschitz we have the Harnack type inequality for x, y ∈ X ,(3.5) e − ε | xy | ≤ ρ ε ( x ) ρ ε ( y ) ≤ e ε | xy | . We remark that by [7, Theorem 5.1] the Harnack inequality (3.5) implies that there is alwaysan ε = ε ( δ ) such that ρ ε is admissible for X for any 0 < ε ≤ ε with constant M = 20.Thus the admissibility hypothesis on ρ ε is not particularly restrictive.We write X ε = X ρ ε for the conformal deformation of X with conformal factor ρ andwrite d ε for the resulting distance on X ε . We write ℓ ε ( γ ) := ℓ ρ ε ( γ ) for the lengths of curvesmeasured in the metric d ε . The properness of X implies that X ε is locally compact. By [14,Theorem 1.4] the metric space X ε is incomplete and unbounded, and bounded geodesics in X are A -uniform curves in X ε . In particular the metric space ( X ε , d ε ) is A -uniform. Wewrite B ε ( x, r ) for the open ball of radius r > x in the metric d ε , and B X ( x, r )for the corresponding 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 [14, Theorem 1.6] there is a canonical identification ι : ∂ ω X → ∂X ε of the boundary of X with respect to ω and the metric boundary ∂X ε of X ε . The correspondence is given byshowing that any sequence { x n } in X converging to a point ξ ∈ ∂ ω X is a Cauchy sequence in X ε converging to a point of ∂X ε . In particular for ξ, ζ ∈ ∂ ω X we can define their distancewith respect to the metric d ε to be d ε ( ι ( ξ ) , ι ( ζ )). We will drop ι from the notation andsimply write d ε ( ξ, ζ ) for this quantity.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 (3.3) 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 [7, 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 three important quantitative results regarding the uniformization X ε from our previous works [14], [13]. The standing assumptions for the rest of this sectionare that X is a proper geodesic δ -hyperbolic space that is K -roughly starlike from a point ω ∈ ∂X , that b : X → R is a Busemann function based at ω , and that for a given ε > ρ ε ( x ) = e − εb ( x ) on X is admissible with constant M . All implied constants willdepend only on δ , K , ε , and M .For this first lemma we set | xy | = ∞ if x = y and either x ∈ ∂ ω X or y ∈ ∂ ω X , and weset | xy | = 0 if x = y ∈ ∂ ω X . Lemma 3.3. [13, Lemma 2.10]
For x, y ∈ X ∪ ∂ ω X we have (3.6) d ε ( x, y ) ≍ e − ε ( x | y ) b min { , | xy |} . In this second lemma we have absorbed the constant ε − on the right in the referenceinto the implied constant. Lemma 3.4. [14, Proposition 4.7]
For x ∈ X we have (3.7) d ε ( x ) ≍ ρ ε ( x ) . The first comparison in the following result is given by the reference. The second com-parison then follows directly from Lemma 3.4.
Lemma 3.5. [13, Lemma 6.2]
There exists < λ < with λ = λ ( δ, K, ε, M ) such that forany x ∈ X and any y, z ∈ B ε ( x, λd ε ( x )) we have that | yz | ≤ and that (3.8) d ε ( y, z ) ≍ d ε ( x ) | yz | ≍ ρ ε ( x ) | yz | . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 11
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 3.6.
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, (3.9) 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 3.4, | 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, 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 (3.9).For the inclusion on the right side of (3.9), 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 3.4, we then have ρ ε ( z ) ≥ C − d ε ( z ) ≥ C − d ε ( x ) ≥ C − ρ ε ( x ) , for a constant C = C ( δ, K, ε, M ) ≥
1. Using this we conclude that r > d ε ( x, y ) ≥ C − ρ ε ( x ) | xy | . Choosing C ∗ to be greater than the constant C on the right side of this inequality, we thenconclude that | xy | < C ∗ rρ ε ( x ) , which gives the right side inclusion in (3.9). (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 having finite diameter in their setting, so we will have totake an approach that is somewhat different. Lemma 3.7.
There is a constant κ = κ ( δ, K, ε, M ) such that if < κ ≤ κ then for every x ∈ ¯ X ε and every r > we can find a ball B ε ( z, κr ) ⊂ B ε ( x, r ) with d ε ( z ) ≥ κr .Proof. Let x ∈ ¯ X ε and r > x ∈ X ε . By Lemma 2.1 wehave b ( X ) = R , which immediately implies that ρ ε ( X ) = (0 , ∞ ). Thus we can find a point z ∈ X such that ρ ε ( z ) = r . Let σ be a geodesic in X that is oriented from x to z , whichwe will assume is parametrized by d ε -arclength. Then σ is an A -uniform curve in X ε with A = A ( δ, K, ε, M ) ≥ 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 ) ≥ C − ρ ε ( z ) = C − r, with C = C ( δ, K, ε, M ) being the implied constant from Lemma 3.4. Thus any κ > κ ≤ C − will work. We then set κ ′ = max { A , C } with C being the impliedconstant of Lemma 3.4. Then the conclusions of the lemma hold for x ∈ X ε , r >
0, and any0 < κ ≤ κ ′ . We set κ = κ ′ .Now let x ∈ ∂X ε and 0 < κ ≤ κ be given. Set r ′ = κr/κ ′ . Then r ′ < r since κ < κ ′ ,so we can find x ′ ∈ X ε sufficiently close to x such that B ε ( x ′ , r ′ ) ⊂ B ε ( x, r ). We then applythe previous claims to x ′ , r ′ , and κ ′ to find z ∈ X ε such that B ε ( z, κ ′ r ′ ) ⊂ B ε ( x ′ , r ′ ) and d ε ( z ) ≥ κ ′ r ′ . It follows that B ε ( z, κr ) = B ε ( z, κ ′ r ′ ) ⊂ B ε ( x ′ , r ′ ) ⊂ B ε ( x, r ) , and d ε ( z ) ≥ κ ′ r ′ = 2 κr. This completes the proof of the lemma. (cid:3)
The conclusion of Lemma 3.7 is closely related to the corkscrew condition for domains inmetric spaces. See [6, Definition 2.4].4.
Global doubling
We will use some standard notation from analysis on metric spaces starting in this section.Some of these notions were previously defined in the introduction; we repeat the definitionsfor reference here. Throughout this paper a metric measure space ( X, d, µ ) will be a tripleconsisting of a metric space (
X, d ) together with a Borel regular measure µ on X such that0 < µ ( B ) < ∞ for all balls B ⊂ X . We make the standard caveat that a ball may bedescribable using more than one center and radius, and for this reason we stipulate that allof our balls come with a fixed choice of center and radius. The measure µ is doubling on X if there is a constant C µ ≥ x ∈ X and any r > µ ( B X ( x, r )) ≤ C µ µ ( B X ( x, r )) . We say that µ is uniformly locally doubling if there is an R > x ∈ X and any 0 < r ≤ R . In this case we will also say that µ is doubling on XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 13 balls of radius at most R . We will frequently make use of the following consequence of thedoubling estimate (4.1): if µ is doubling on balls of radius at most R with constant C µ and0 < r ≤ R ≤ R then(4.2) 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 (4.1) and then also 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 4.1. [2, Proposition 3.2]
Let ( X, d ) be a geodesic metric space and let µ be ameasure 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 , with doublingconstant 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 that is K -roughly starlike from a point ω ∈ ∂X and let b be a Busemann function on X based at ω . We let ε > ρ ε is admissible 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 Borelregular measure on X such that 0 < µ ( B ) < ∞ forall balls B ⊂ X . We will assume that there is an R > µ is doubling on balls ofradius at most R with doubling constant C µ .For each β > µ β on X by(4.3) dµ β ( x ) = ρ βε ( x ) dµ ( x ) = e − βεb ( x ) dµ ( x ) , for x ∈ X . We will consider µ β as a measure on X ε and extend it to the completion ¯ X ε bysetting µ β ( ∂X ε ) = 0. In this section we will be establishing a criterion (Definition 4.6) for µ β to be doubling on ¯ X ε . Remark . Our notation for µ β is different from the notation used in the analogous settingin [2]; our µ β corresponds to µ βε in [2]. We suppress the dependence on ε in the notationfor µ β because ε will be considered to be fixed in all settings that we consider.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 Proposition4.1. When we do this we will also need to increase C µ by a corresponding amount dependingonly on the uniformization data. 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. This distinction will only be important in the proof of Proposition 4.12, which isn’tneeded for any of our primary claims. Our first claim corresponds to [2, Lemma 4.5]. It provides an important estimate on themeasure of subWhitney balls in X ε . Lemma 4.4.
Let x ∈ X and < r ≤ d ε ( x ) . Then µ β ( B ε ( x, r )) ≍ ρ βε ( 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 3.4 we have for all y ∈ B ε ( x, r ),(4.4) ρ βε ( y ) = ρ ε ( y ) β ≍ d ε ( y ) β ≍ d ε ( x ) β ≍ ρ βε ( x ) , with the comparison d ε ( y ) ≍ d ε ( x ) following from the condition on r . Applying Lemma3.6 and the chain of comparisons (4.4), we conclude that µ β ( B ε ( x, r )) ≍ ρ βε ( x ) µ ( B ε ( x, r )) . ρ βε ( 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 3.6. A similar argument showsthat µ β ( B ε ( x, r )) & ρ βε ( x ) µ (cid:18) B X (cid:18) x, C − ∗ rρ ε ( x ) (cid:19)(cid:19) . We thus conclude that(4.5) ρ βε ( x ) µ (cid:18) B X (cid:18) x, C − ∗ rρ ε ( x ) (cid:19)(cid:19) . µ β ( B ε ( x, r )) . ρ βε ( x ) µ (cid:18) B X (cid:18) x, C ∗ rρ ε ( x ) (cid:19)(cid:19) The condition on r implies that(4.6) rρ ε ( x ) ≤ d ε ( x ) ρ ε ( x ) ≤ C, with C depending only on the uniformization data by Lemma 3.4. By Proposition 4.1 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 (4.6) and theconstant C ∗ in Lemma 3.6. Then the comparison (4.2) 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 (4.5) proves the lemma. (cid:3)
By combining Lemma 4.4 with Lemma 3.7 we obtain the following estimate for µ β ( B ε ( x, r ))when 0 < r ≤ d ε ( x ). Below we let κ = κ ( δ, K, ε, M ) be defined as in Lemma 3.7. Lemma 4.5.
Let x ∈ X and < r ≤ d ε ( x ) . Let < κ ≤ κ and z ∈ X be given such that B ε ( z, κr ) ⊂ B ε ( x, r ) and d ε ( z ) ≥ κr . Then µ β ( B ε ( x, r )) ≍ µ β ( B ε ( z, κr )) , with comparison constants depending only on the data and κ .Proof. By Lemmas 3.4 and 3.6 we have(4.7) | 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 3.6. 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 3.4 that ρ ε ( z ) ≍ ρ ε ( x ) with comparison constantdepending only on the uniformization data. Since d ε ( z ) ≥ κr , we have by Lemma 3.4 that(4.8) 1 & κrρ ε ( z ) ≍ rρ ε ( z ) ≍ rρ ε ( x ) ≍ C ∗ rρ ε ( x ) , with all implied constants depending only on the uniformization data and κ . We can thusapply Proposition 4.1 to conclude that we can assume that µ is doubling on balls of radiusat most any of the terms appearing in (4.8), at the cost of increasing the doubling constantof µ by an amount depending only on the uniformization data and κ . It follows that µ (cid:18) B X (cid:18) z, κrρ 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 and κ . The third comparison abovefollows from the fact that z ∈ B X (cid:16) x, C ∗ rρ ε ( x ) (cid:17) by (4.7). Since the comparison ρ βε ( z ) ≍ ρ βε ( x )follows from the comparison ρ ε ( z ) ≍ ρ ε ( x ) (with comparison constants depending onlyon the uniformization data), applying Lemma 4.4 to B ε ( z, κr ) and B ε ( x, r ) (note that κr ≤ d ε ( z ) by assumption) then gives µ β ( B ε ( z, κr )) ≍ µ β ( B ε ( x, r )) , with comparison constants depending only on the data and κ . (cid:3) In order to obtain doubling of µ β on ¯ X ε , we need to be able to extend Lemma 4.5 to cover x ∈ ¯ X ε and all r >
0. The following definition axiomatizes this condition. We will use thistechnical definition to verify doubling of the lifted measures in Section 7. We let κ > z as specified below is given by Lemma 3.7. Werecall that the measure µ β on X ε is extended to the completion ¯ X ε by setting µ β ( ∂X ε ) = 0. Definition 4.6.
The measure µ β is ∂ -controlled if for each 0 < κ ≤ κ there exists a constant C ∂ ( κ ) ≥ ξ ∈ ∂X ε , r >
0, and z ∈ X such that B ε ( z, κr ) ⊂ B ε ( ξ, r ) and d ε ( z ) ≥ κr , we have(4.9) µ β ( B ε ( ξ, r )) ≍ C ∂ ( κ ) µ β ( B ε ( z, κr )) . Observe that the lower bound in (4.9) follows trivially from the inclusion B ε ( z, κr ) ⊂ B ε ( ξ, r ). Thus it is only the upper bound that is of interest in (4.9). To motivate Definition4.6, we note that being ∂ -controlled is necessary for µ β to be doubling on ¯ X ε . Proposition 4.7.
Suppose that µ β is doubling on ¯ X ε with constant C µ β . Then µ β is ∂ -controlled with C ∂ ( κ ) depending only on κ and C µ β for each < κ ≤ κ .Proof. Let 0 < κ ≤ κ , ξ ∈ ∂X ε , r >
0, and z ∈ X be given as in Definition 4.6. Then B ε ( ξ, r ) ⊂ B ε ( z, r ) since z ∈ B ε ( ξ, r ), which implies that µ β ( B ε ( z, κr )) ≤ µ β ( B ε ( ξ, r )) ≤ µ β ( B ε ( z, r )) . By (4.2) applied to µ β we conclude that µ β ( B ε ( z, κr )) ≍ µ β ( B ε ( ξ, r )) with comparisonconstant depending only on κ and C µ β . (cid:3) When µ β is ∂ -controlled we can improve Lemma 4.5 to hold for all x ∈ ¯ X ε and r > Lemma 4.8.
Suppose that µ β is ∂ -controlled. Let x ∈ ¯ X ε , r > , < κ ≤ κ and z ∈ X begiven such that B ε ( z, κr ) ⊂ B ε ( x, r ) and d ε ( z ) ≥ κr . Then (4.10) µ β ( B ε ( x, r )) ≍ µ β ( B ε ( z, κr )) , with comparison constant depending only on the data, κ , and C ∂ ( κ/ .Proof. The lower bound µ β ( B ε ( x, r )) ≥ µ β ( B ε ( z, κr )) follows from the inclusion B ε ( z, κr ) ⊂ B ε ( x, r ), so we only need to establish the upper bound in (4.10). By Lemma 4.5 it sufficesto consider the case x ∈ ¯ X ε , r > d ε ( x ). We can then find ξ ∈ ∂X ε such that B ε ( x, r ) ⊂ B ε ( ξ, r ). Applying (4.9) with κ replacing κ , we conclude that µ β ( B ε ( x, r )) ≤ µ β ( B ε ( ξ, r )) ≍ C ∂ ( κ/ µ β ( B ε ( z, κr )) . This gives the desired upper bound in the case r > d ε ( x ) with comparison constant C ∂ ( κ/ (cid:3) We will now show that µ β is doubling on ¯ X ε when it is ∂ -controlled, with doublingconstant depending only on the data and the particular constants C ∂ ( κ /
6) and C ∂ ( κ / κ = κ / κ = κ / Proposition 4.9.
Suppose that µ β is ∂ -controlled. Then µ β is doubling on ¯ X ε with doublingconstant depending only on the data and max { C ∂ ( κ / , C ∂ ( κ / } .Proof. For x ∈ ¯ X ε we use Lemma 3.7 to fix κ = κ > z ∈ X such that the ball B ε ( z, κr ) is contained in B ε ( x, r ) (and therefore also in B ε ( x, r )) and d ε ( z ) ≥ κr . We set κ ′ = κ . Since d ε ( z ) ≥ κr = 2 κ ′ · r , we can apply Lemma 4.8 twice to obtain µ β ( B ε ( x, r )) ≍ µ β ( B ε ( z, κ ′ r )) = µ β ( B ε ( z, κr )) ≍ µ β ( B ε ( x, r )) , with implied constants depending only on the data, κ , and max { C ∂ ( κ / , C ∂ ( κ / } ,which means they only depend on the data and max { C ∂ ( κ / , C ∂ ( κ / } since κ isdetermined by the uniformization data. (cid:3) In practice it is often easier to verify the following condition when attempting to showthat µ β is ∂ -controlled. Lemma 4.10.
Suppose that for some < κ ≤ κ there is a constant C ′ ∂ ( κ ) ≥ such that,for any ξ ∈ ∂X ε , r > , and z ∈ X with B ε ( z, κr ) ⊂ B ε ( ξ, r ) and d ε ( z ) ≥ κr , we have (4.11) µ β ( B ε ( ξ, r )) . C ′ ∂ ( κ ) ρ βε ( z ) µ ( B X ( z, . Then the comparison (4.9) holds for this value of κ with comparison constant C ∂ ( κ ) depend-ing only on the data, κ , and C ′ ∂ ( κ ) .Consequently if (4.11) holds for all < κ ≤ κ then µ β is ∂ -controlled with constants C ∂ ( κ ) depending only on the data, κ , and C ′ ∂ ( κ ) .Proof. As remarked after Definition 4.6, it suffices to verify the upper bound in (4.9). Since κr ≤ d ε ( z ), we can apply Lemma 4.4 to obtain µ β ( B ε ( z, κr )) ≍ ρ βε ( z ) µ (cid:18) B X (cid:18) z, κrρ ε ( z ) (cid:19)(cid:19) , XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 17 with comparison constants depending only on the data. We then observe by Lemma 3.4and the fact that d ε ( z ) < r since z ∈ B ε ( ξ, r ),(4.12) κC − ≤ κd ε ( z ) ρ ε ( z ) ≤ κrρ ε ( z ) ≤ d ε ( z )2 ρ ε ( z ) ≤ C, for a constant C = C ( δ, K, ε, M ) ≥ C µ of µ by an amountdepending only on the data, assume that µ is doubling on balls of radius at most C . Wethen conclude that µ (cid:0) B X (cid:0) z, C − κ (cid:1)(cid:1) ≍ µ ( B X ( z, C )) ≍ µ ( B X ( z, , with comparison constants depending only on the data and κ , which implies by (4.12) that µ (cid:18) B X (cid:18) z, κrρ ε ( z ) (cid:19)(cid:19) ≍ µ ( B X ( z, , again with comparison constants depending only on the data and κ . We thus conclude that µ β ( B ε ( z, κr )) ≍ ρ βε ( z ) µ ( B X ( z, , with comparison constant depending only on the data and κ . Our assumption then impliesthat µ β ( B ε ( ξ, r )) . C ′ ∂ ( κ ) ρ βε ( z ) µ ( B X ( z, ≍ µ β ( B ε ( z, κr )) . We conclude that the comparison (4.9) holds with constant C ∂ ( κ ) depending only on thedata, κ , and C ′ ∂ ( κ ). (cid:3) The next result will not be used in the main theorems of this paper, since we will beverifying the doubling property for measures on hyperbolic fillings in Section 7 in a differentfashion. We will show, 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 of Proposition 4.1. Lemma 4.11. [2, Lemma 3.5]
Let ( X, d, µ ) be a geodesic metric measure space such that µ is doubling on balls of radius at most R with constant C µ . Let n ∈ N be a given integer.(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 . Proposition 4.12.
There is β = β ( δ, K, ε, M, R , C µ ) > such that if β ≥ β then µ β isdoubling on ¯ X ε with constant C µ β depending only on the data.Proof. By the proof of Proposition 4.9 it suffices to verify that the comparison (4.9) holdsfor κ = κ / κ = κ /
12. By Lemma 4.10 it suffices to show that the inequality (4.11)holds for κ = κ / κ = κ /
12. Let ξ ∈ ∂X ε and r > κ > z ∈ X are given such that B ε ( z, κr ) ⊂ B ε ( ξ, r ) and d ε ( z ) ≥ κr , where κ = κ /c and c = 6 or c = 12. We define for n ≥ A n = { x ∈ B ε ( ξ, r ) ∩ X : e − n r ≤ d ε ( x ) < e − n r } . Since x ∈ B ε ( ξ, r ) implies that d ε ( x ) < r , we have B ε ( ξ, r ) ∩ X = 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 3.3 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 )) κe − n r ≤ c e n κ . e n , with implied constant depending only on δ , K , ε , and M , since κ = κ ( δ, K, ε, M ) and c ≤
12. We then conclude that ε | xz | ≤ n + c ′ , with c ′ = c ′ ( δ, K, ε, M ) ≥
0. Sincethis inequality trivially holds with c ′ = ε when | xz | <
1, we in fact obtain the inequality ε | xz | ≤ n + c ′ in both cases. Since n ≥
1, we thus obtain that | xz | ≤ ε − (1 + c ′ ) n. We set c ∗ = ε − (1 + c ′ ), observing that c ∗ = c ∗ ( δ, K, ε, M ). Then | xz | ≤ c ∗ n for x ∈ A n .We now apply Proposition 4.1 to ensure that µ is doubling on balls of radius at most R = max { R + c ∗ , } . The doubling constant C ′ µ for µ on balls of radius at most R thendepends only on δ , K , ε , M , R , and C µ . In particular C ′ µ does not depend on β . Applying(2) of Lemma 4.11, we cover A n ⊂ B X ( z, c ∗ n ) with N n . e αn many balls B n,j of radius R ,where α = α ( δ, K, ε, M, R , C µ ) = 74 log C ′ µ . Hence α also does not depend on β . We set β := 2 α and assume that β ≥ β .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 + c ∗ n < R n, since n ≥
1. Applying (1) of Lemma 4.11 then gives that µ ( B n,j ) ≤ ( C ′ µ ) n µ ( B X ( z, R )) ≤ e αn µ ( B X ( z, R )) , Since d ε ( z ) ≍ κ r , we obtain that ρ βε ( z ) ≍ d ε ( z ) β ≍ r β , with comparison constants depending only on the uniformization data. Likewise for x ∈ A n ,(4.13) ρ βε ( x ) ≍ d ε ( x ) β ≍ ( e − n r ) β , again with comparison constants depending only on the uniformization data. The Harnackinequality (3.5) implies that ρ βε ( y ) ≍ C ρ βε ( x n,j ) for each y ∈ B n,j , with C depending onlyon the data (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 (4.13) that ρ βε ( x n,j ) ≍ ( e − n r ) β ,with comparison constant depending only on the uniformization data. Thus we conclude XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 19 that µ β ( A n ∩ B n,j ) ≤ µ β ( B n,j ) ≍ ρ β ( x n,j ) µ ( B n,j ) . ( e − n r ) β µ ( B n,j ) . e − βn ρ β ( z ) µ ( B X ( z, R )) , with implied constants depending only on the data. By our restriction β ≥ β = 2 α , weconclude that µ β ( A n ∩ B n,j ) . e − αn ρ β ( z ) µ ( B X ( z, R )) . It then follows from this inequality and the bound N n . e αn that µ β ( B ε ( ξ, r )) ≤ ∞ X n =1 N n X j =1 µ β ( A n ∩ B n,j ) . ρ β ( z ) µ ( B X ( z, R )) ∞ X n =1 N n e − αn . ρ β ( z ) µ ( B X ( z, R )) ∞ X n =1 e − αn . ρ β ( z ) µ ( B X ( z, R )) . with comparison constants depending only on the data, with the final inequality followingby summing the geometric series. Finally, since R depends only on the data and R ≥ µ ( B X ( z, R )) ≍ µ ( B X ( z, µ β ( B ε ( ξ, r )) . ρ β ( z ) µ ( B X ( z, , with implied constant depending only on the data. As noted at the beginning of this proof,this implies the desired comparisons (4.9) for κ = κ / κ = κ /
12 by Lemma 4.10. (cid:3) Global Poincar´e inequality
Let (
X, d, µ ) be a metric measure space. For a measurable subset E ⊂ X satisfying0 < µ ( E ) < ∞ and a function u that is µ -integrable over E we write u E = − Z E u dµ = 1 µ ( E ) Z E u dµ for the mean value of u over E . We record the following simple lemma for use later. Lemma 5.1. [1, Lemma 4.17]
Let u : X → R be integrable, let ≤ p < ∞ , let α ∈ R , andlet 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 Let u : X → R be given. We recall from the introduction that 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 ≥ λ ≥ X supports a p -Poincar´e inequality if there 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(5.1) − 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 . For more details on Poincar´e inequalitieswe refer to Section 8.For this section we carry over the same standing hypotheses and notation as discussedat the start of Section 4. We will assume in addition that we are given p ≥ X equipped with the uniformly locally doubling measure µ supports a p -Poincar´e inequality on balls of radius at most R , where R is the sameradius up to which µ is doubling on X . We will also assume that µ β is doubling on ¯ X ε forsome constant C µ β . We will show under these hypotheses that the metric 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 of µ , the global doubling of µ β , and the uniformlylocal p -Poincar´e inequality on X .The proof splits into two steps. In the first step we show that (5.1) holds on sufficientlysmall subWhitney balls in X ε . The proof is essentially identical to [2, Lemma 6.1]. In thestatement and proof of Lemma 5.2 “the data” refers to the uniformization data and theconstants R , C µ , p , λ , and C PI . For Lemma 5.1 we do not need to assume that µ β isdoubling. Lemma 5.2.
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 (5.1) for µ β holds onthe ball B ε ( x, r ) with dilation 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 c > C ∗ be the constant of Lemma 3.6. We choose c > c C ∗ ≤ . Weconclude by applying Lemma 3.6 twice that(5.2) 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 (4.4) we see that for all y ∈ ˆ λB ε we have ρ βε ( y ) ≍ ρ βε ( 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 3.4 that C ∗ rρ ε ( x ) ≤ C ∗ c d ε ( x ) ρ ε ( x ) ≤ R . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 21
Thus the p -Poincar´e inequality (5.1) (for µ ) holds on B . Since ρ βε ( y ) ≍ ρ βε ( x ) on ˆ λB ε withcomparison constant depending only on the uniformization data (by (4.4)) we have that(5.3) µ β ( B ) ≍ ρ βε ( 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 (5.2), the measure comparison (5.3), and the p -Poincar´e inequalityfor 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 5.1 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 5.3. [2, Proposition 6.3]
Let Ω be an A -uniform metric space equipped with aglobally doubling measure ν such that there is a constant < c < for which the p -Poincar´einequality (5.1) 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 , 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 . Below “the data” includes allthe constants from Lemma 5.2 as well as the doubling constant C µ β for µ β . Proposition 5.4.
The metric measure spaces ( X ε , d ε , µ β ) and ( ¯ X ε , d ε , µ β ) support a global p -Poincar´e inequality with dilation constant and constant C ∗ PI depending only on the data.Proof. By Lemma 5.2 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 metric spacewith 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 5.3 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 (5.1) in fact holds with dilationconstant 1, with constant C ∗ PI depending only on the data [19, Theorem 4.18].By [21, Lemma 8.2.3] it follows that the completion ( ¯ X ε , d ε , µ β ) (with µ β ( ∂X ε ) = 0) alsosupports a p -Poincar´e inequality with constants depending only on the constants for the p -Poincar´e inequality on X ε and the doubling constant of µ β . Since ¯ X ε is also geodesic itfollows by the same reasoning [19, Theorem 4.18] that we can take the dilation constant tobe 1 in this case as well. (cid:3) Hyperbolic fillings
Let (
Z, d ) be a metric space and let 0 < a < τ > τ > (cid:26) , − a (cid:27) . We write B Z ( z, r ) for the ball of radius r centered at z in Z . We construct a hyperbolicfilling X of Z as in [14]. A subset S ⊂ Z is r -separated for a given r > x, y ∈ S we have d ( x, y ) ≥ r . For each n ∈ Z we select a maximal a n -separated subset S n of Z . Thenfor each n ∈ Z the balls B Z ( z, a n ), z ∈ S n , cover Z .The vertex set of X has the form V = [ n ∈ Z V n , V n = { ( z, n ) : z ∈ S n } . To each vertex v = ( z, n ) we associate the dilated ball B ( v ) := B Z ( z, τ a n ). We define aprojection π : V → Z by setting π ( z, n ) = z and define the height function h : V → Z by h ( z, n ) = n . For a vertex v ∈ V we will sometimes write v in place of π ( v ) and consider v both as a point of Z and a vertex of X , except in places where this could cause confusion.We place an edge between two vertices v and w if | h ( v ) − h ( w ) | ≤ B ( v ) ∩ B ( w ) = ∅ .For vertices v, w ∈ V we will write v ∼ w if there is an edge joining v to w . We write E forthe set of edges in X . The resulting graph X is connected by [14, Proposition 5.5]. We give X the geodesic metric in which all edges have unit length. We will write B X ( x, r ) for theball of radius r > x ∈ X , as in the previous sections. We extend the heightfunction piecewise linearly to the edges of X to define a 1-Lipschitz function h : X → R .We say that an edge e is vertical if it connects two vertices of different heights and horizontal if it connects two vertices of the same height. A vertical edge path in X is asequence of edges joining a sequence of vertices { v k } with h ( v k +1 ) = h ( v k ) + 1 for each k or h ( v k +1 ) = h ( v k ) − k . In the first case we say that the edge path is ascending and in the second case we say that the edge path is descending . We observe vertical edgepaths are always geodesics in X since edges in X can only join two vertices of the same oradjacent heights. Thus we will also refer to vertical edge paths as vertical geodesics . Thefollowing is an easy observation from the construction of X . Lemma 6.1. [14, Lemma 5.3]
Let v, w ∈ V with h ( v ) = h ( w ) and B ( v ) ∩ B ( w ) = ∅ . Thenthere is a vertical geodesic connecting v to w . The metric space Z is doubling if there is an integer D ≥ z ∈ Z and r > r -separated subset of B Z ( z, r ) has at most D points. We say that the graph X has bounded degree if there is an integer N ≥ N other vertices. Note that if X has bounded degree then it is proper, i.e.,closed balls in X are compact. These two concepts are closely linked to one another, as thefollowing proposition shows. Proposition 6.2.
The metric space Z is doubling if and only if the hyperbolic filling X hasbounded degree, and this equivalence is quantitative in the doubling constant, vertex degree, a , and τ . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 23
Proof.
We give a proof that closely follows the proof of an analogous proposition in the workof Buyalo-Schroeder [15, Proposition 8.3.3]. We first show that X has bounded degree if Z is doubling. It’s an easy standard fact that if Z is doubling with constant D then there isa control function Λ : [1 , ∞ ) → N , quantitative in D , such that for each r > sr contains at most Λ( s ) points that are r -separated (see [15, Exercise 8.3.1] ). Thisfollows by simply applying the doubling condition repeatedly across multiple scales.Let v ∈ V n be any vertex and consider the associated ball B ( v ). For each m ∈ Z welet S m ( v ) ⊂ S m denote the set of points z ∈ S m such that the associated vertex w ∈ V m satisfies v ∼ w . Then we can only have S m ( v ) = ∅ when | m − n | ≤
1. If w ∈ S m ( v ) then B ( v ) ∩ B ( w ) = ∅ and m ≤ n −
1. This implies that d ( π ( v ) , π ( w )) < τ a n − . Thus S n − ( v ), S n ( v ), and S n +1 ( v ) each form an a n +1 -separated set inside of the ball B ( π ( v ) , τ a n − ) andthus each have cardinality bounded by Λ(2 τ a − ). We conclude that X has bounded degreewith degree bound N = 3Λ(2 τ a − ).Let’s now assume that X has bounded degree, so that each vertex of X is connectedto at most N other vertices. Consider a ball B Z ( z, r ) in Z . Let k ∈ Z be such that a k +1 < r ≤ a k . Let l ∈ N be the minimal integer such that a l − ≤ . Then any r -separatedsubset of B Z ( z, r ) is also a k + l -separated. Let v ∈ V k be such that d ( z, π ( v )) < a k . Thenfor y ∈ B Z ( z, r ) we have d ( y, π ( v )) ≤ d ( y, z ) + d ( z, π ( v )) < a k < τ a k , since τ >
3, which implies that B Z ( z, r ) ⊆ B ( v ). It thus suffices to show that there is auniform bound on the size of an a k + l -separated subset of B ( v ), quantitative in N , a , l , and τ (note l is quantitative in a ).If { z n } is an a k + l -separated subset of B ( v ) then the balls B Z ( z n , a k +2 l ) are all disjointsince a l < . We select for each z n a corresponding vertex v n ∈ V k +2 l such that π ( v n ) ∈ B Z ( z n , a k +2 l ); these vertices are all distinct since these balls are disjoint. We then have B Z ( z n , a k +2 l ) ⊂ B Z ( π ( v n ) , a k +2 l ) ⊂ B ( v n ) , since τ >
3, which implies that the vertex v n then satisfies B ( v n ) ∩ B ( v ) = ∅ . It thus sufficesto produce a uniform bound on the number of vertices w ∈ V k +2 l such that B ( w ) ∩ B ( v ) = ∅ .Given such a vertex w , since B ( w ) ∩ B ( v ) = ∅ and l ≥
1, by Lemma 6.1 we can find a verticalgeodesic from w to v of length 2 l . We conclude that any vertex w ∈ V k +2 l with B ( w ) ∩ B ( v )is joined to v by a vertical geodesic of length 2 l . Since the number of vertices joined to v by a vertical geodesic of length 2 l is at most N l , this produces our desired bound. (cid:3) We will assume for the rest of this section that Z is a complete doubling metric space, fromwhich it follows that X is a proper geodesic metric space. Note that the doubling conditionon Z implies that bounded subsets of Z are totally bounded, from which it follows that anyclosed and bounded subset of Z is compact. We conclude in particular that Z is proper.The following consequence of the doubling condition will be used frequently in subsequentsections. Lemma 6.3.
Let Z be doubling with doubling constant D . Let S be an r -separated subsetof Z for a given r > . Then for any τ ≥ and z ∈ Z we have that z ∈ B Z ( x, τ r ) for atmost D l +1 points x ∈ S , where l is the minimal integer such that l ≥ τ .Proof. Suppose that z ∈ B Z ( x, τ r ) for some x ∈ S . Then y ∈ B Z ( x, τ r ) for any other y ∈ S such that z ∈ B Z ( y, τ r ). Thus S ∩ B Z ( x, τ r ) defines an r -separated subset of theball B Z ( x, τ r ), which is also an r -separated subset of the ball B Z ( x, l +1 r ). The doublingproperty then implies that the cardinality of S ∩ B Z ( x, τ r ) is bounded above by D l +1 . (cid:3) Throughout the rest of this section all implied constants are considered to only dependon a and τ . By [14, Proposition 5.9] we have that the hyperbolic filling X is δ -hyperbolicwith δ = δ ( a, τ ).A vertical geodesic γ in X is anchored at a point z ∈ Z if for each vertex v on γ we havethat z ∈ B ( π ( v ) , a h ( v ) ). Note this implies that z ∈ B ( v ). If we do not need to mention thepoint z then we will just say that γ is anchored . The following simple lemma shows that forany z ∈ Z we can find a geodesic line anchored at z . Lemma 6.4.
For any z ∈ Z there is a vertical geodesic line γ : R → X anchored at z . If v ∈ V satisfies z ∈ B ( π ( v ) , a h ( v ) ) then we can choose γ such that v ∈ γ .Proof. For each n ∈ Z the balls B ( π ( v ) , a n ) for v ∈ V n cover Z . Thus for each n we canchoose a vertex v n ∈ Z such that z ∈ B ( π ( v n ) , a n ). Then z ∈ B ( v n ) as well. It followsthat B ( v n ) ∩ B ( v n +1 ) = ∅ for each n ∈ Z , i.e., v n ∼ v n +1 . We can then find a verticalgeodesic γ : R → X such that v n ∈ γ for each n . If v ∈ V is a given vertex such that z ∈ B ( π ( v ) , a h ( v ) ) then we can choose v h ( v ) = v in this construction to guarantee that v ∈ γ . (cid:3) Remark . The definition of anchored geodesics in [14] is less restrictive, requiring only that z ∈ B ( π ( v ) , τ a h ( v ) ) for each v ∈ γ instead. This was because we were not assuming that Z was complete in that paper. We will use the more restrictive inclusion z ∈ B ( π ( v ) , a h ( v ) ) hereinstead, with the understanding that all of the claims proved regarding anchored geodesicsin [14] hold for these geodesics in particular.By [14, Lemma 5.11] all descending anchored geodesic rays in X are at a bounded distancefrom one another and therefore define a common point ω ∈ ∂X . By [14, Proposition 5.13]we have a canonical identification ∂ ω X ∼ = Z that can be realized by identifying a point z ∈ Z with the collection of ascending geodesic rays anchored at z . Under this identification themetric d on Z defines a visual metric on ∂ ω X with parameter q = − log a . Furthermore by[14, Lemma 6.1] we have that X is -roughly starlike from ω .By [14, Lemma 5.12] there is a Busemann function b based at ω such that the heightfunction h satisfies h . = b . Thus the height function h can be thought of as a Busemannfunction on X based at ω up to an additive error of 3. We define the Gromov product basedat h by ( x | y ) h = 12 ( h ( x ) + h ( y ) − | xy | ) , for x, y ∈ X . We then observe that ( x | y ) h . = ( x | y ) b as well. Since h is 1-Lipschitz we alsohave ( x | y ) h ≤ min { h ( x ) , h ( y ) } , for x, y ∈ X . The Gromov product based at h is then extended to ∂ ω X by the sameformulas (2.6) and (2.8) as were used for b , and the same estimates (2.7) and (2.9) holdfor this extension with h replacing b . We have the following key estimate relating Gromovproducts based at h to the distance on Z . We recall that all implied constants depend onlyon a and τ . We also recall that X is δ -hyperbolic with δ = δ ( a, τ ). Lemma 6.6. [14, Lemma 5.6]
For v, w ∈ V we have a ( v | w ) h ≍ d ( v, w ) + max { a h ( v ) , a h ( w ) } . The following observation will also be useful.
XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 25
Lemma 6.7.
For any vertex v ∈ V we have ( v | π ( v )) h . = h ( v ) . Proof.
By Lemma 6.4 we can find an ascending geodesic ray γ : [0 , ∞ ) → X startingfrom v that is anchored at π ( v ). Then a straightforward calculation shows that we have( γ ( t ) | v ) h = h ( v ) for all t ≥
0. Since the sequence { γ ( n ) } ∞ n =0 converges to π ( v ) ∈ ∂ ω X ∼ = Z ,the conclusion then follows from inequality (2.7). (cid:3) We define a density ρ : X → (0 , ∞ ) by ρ ( x ) = a h ( x ) and write X ρ for the conformaldeformation of X with conformal factor ρ . We write d ρ for the metric on X ρ . By [14,Theorem 1.9] the density ρ is admissible for X with constant M = M ( a, τ ) and boundedgeodesics in X are A -uniform curves in X ρ with A = A ( a, τ ). In particular X ρ is an A -uniform metric space. Furthermore we have a canonical L -biLipschitz identification of ∂X ρ with Z , with L = L ( a, τ ). Thus after a biLipschitz change of metric on Z we can assumethat Z is isometrically identified with ∂X ρ . We will make this biLipschitz change of metricand thus consider Z as an isometrically embedded subset of ¯ X ρ with Z = ∂X ρ . We will thenuse the notation Z and ∂X ρ interchangeably for Z , with the choice of notation dependingon the context. For x ∈ ¯ X ρ and r > B ρ ( x, r ) for the ball of radius r centeredat x in the metric d ρ .Set ε = − log a . Let b be a Busemann function based at ω such that b . = h and define adensity ρ ε : X → (0 , ∞ ) on X by(6.2) ρ ε ( x ) = e − εb ( x ) = a b ( x ) . Then X ε is biLipschitz by the identity map to X ρ with biLipschitz constant a − . In par-ticular ρ ε is also an admissible density on X with constant M = M ( a, τ ). We thus deducethat the results of Section 3 also hold for X ρ . In particular we have the following two keylemmas, in which the implied constants depend only on a and τ . We recall below that wedefine | xy | = ∞ if x = y and either x ∈ ∂ ω X or y ∈ ∂ ω X , and set | xy | = 0 if x = y ∈ ∂ ω X .Recall also that we have canonically identified ∂ ω X with Z . Lemma 6.8.
Let x, y ∈ X ∪ ∂ ω X . Then we have d ρ ( x, y ) ≍ a ( x | y ) h min { , | xy |} . Lemma 6.9.
For x ∈ X we have d ρ ( x ) ≍ a h ( x ) . In connection with Lemma 6.9 the following observation will be useful.
Lemma 6.10.
For v ∈ V we have d ρ ( v, π ( v )) ≍ d ρ ( v ) ≍ a h ( v ) . Proof.
We trivially have d ρ ( v, π ( v )) ≥ d ρ ( v ). On the other hand, by Lemma 6.4 we can findan ascending vertical geodesic ray γ : [0 , ∞ ) → X starting at v and anchored at π ( v ). Astraightforward computation shows that ℓ ρ ( γ ) = Z ∞ a t + h ( v ) dt . a h ( v ) , from which it follows that d ρ ( v, π ( v )) . a h ( v ) . The conclusion of the lemma then followsfrom Lemma 6.9. (cid:3) We now introduce a concept inspired by work of Lindquist [23, Definition 4.10]. Ourdefinition is somewhat different from the one given there. For this definition we recall thatballs are always considered to have a fixed center and radius, even if a particular subset canbe described as a ball in multiple different ways. The quantity 2 r below comes from theupper bound diam( B ) ≤ r . Definition 6.11.
Let B = B Z ( z, r ) be any ball in Z . The hull H B ⊂ X ρ of B in X ρ isthe union H B = S ¯ B X ( v, ) over all vertices v ∈ V such that a h ( v ) ≤ r and B ( v ) ∩ B = ∅ .We consider H B as being equipped with the uniformized metric d ρ . For n ∈ Z we write H Bn = H B ∩ V n for the set of vertices in H B at height n .Since balls are considered to come assigned with a center and radius, it may be thecase for two balls B and B ′ in Z with different centers and radii that B = B ′ as sets but H B = H B ′ . By construction each vertex v ∈ H B has the property that a h ( v ) ≤ r and B ( v ) ∩ B = ∅ , and for each x ∈ H B there is a vertex v ∈ H B such that | xv | ≤ . An edge e in X satisfies e ⊂ H B if and only if the endpoints of e both belong to H B . For each n ∈ Z such that a n ≤ r we have by construction that the balls B ( v ) for v ∈ H Bn cover B . In thispaper we will primarily be using hulls of balls in Z as a convenient approximation to ballsin ¯ X ρ centered at points of Z . We first show that the metric boundary of H B coincideswith the closure ¯ B of B in Z . Lemma 6.12.
Let B ⊂ Z be any ball and let H B ⊂ X ρ be its hull. Then ∂H B = H B \ H B = ¯ B ⊂ Z. Proof.
Let r = r ( B ). Let { x n } ⊂ H B be any sequence for which there is some y ∈ ¯ X ρ \ H B such that d ρ ( x n , y ) →
0. If y ∈ X ρ then, since the metric d ρ is locally biLipschitz to thehyperbolic metric on X , we must have | x n y | → B X ( v, )for v ∈ V cover X , we can find a vertex v such that y ∈ ¯ B X ( v, ). If y ∈ B X ( v, ) then wemust have x n ∈ B X ( v, ) for n sufficiently large. Since x n ∈ H B this implies that v ∈ H B ,from which it follows that y ∈ H B , contradicting our assumptions. If | yv | = then we let w be the other vertex on the edge containing y . Then for n sufficiently large we must haveeither x n ∈ ¯ B X ( v, ) or x n ∈ ¯ B X ( w, ). Thus we must have either v ∈ H B or w ∈ H B ,both of which imply that y ∈ H B . This again contradicts our assumptions.Thus we must have y ∈ Z = ∂X ρ . Then { x n } converges to a point of ∂X ρ and therefore h ( x n ) → ∞ as n → ∞ . By the definition of H B we can find a sequence of vertices { v n } ⊂ H B such that | x n v n | ≤ . By Lemma 6.8 it follows that d ρ ( x n , v n ) →
0, so wealso have d ρ ( v n , y ) →
0. Set z n = π ( v n ). Then d ρ ( z n , v n ) . a h ( v n ) by Lemma 6.10 andtherefore d ρ ( z n , v n ) →
0. Thus d ρ ( z n , y ) →
0. But since B ∩ B ( v n ) = ∅ , we can find points y n ∈ B ∩ B ( v n ) for each n that satisfy d ( y n , z n ) < τ a h ( v n ) . This implies that d ( y n , y ) → y ∈ ¯ B since y n ∈ B for each n .We conclude that ∂H B ⊂ ¯ B . To obtain equality, let y ∈ ¯ B be any point and let γ be anascending vertical geodesic ray anchored at y as constructed in Lemma 6.4 starting from avertex v ∈ V . Let { v n } n ≥ be the sequence of vertices on γ with h ( v n ) = n . For each n we then have y ∈ B ( v n ) by construction and therefore B ( v n ) ∩ B = ∅ by the definition ofthe closure ¯ B . For n sufficiently large we will have that a n ≤ r and therefore v n ∈ H B .Then d ρ ( v n , y ) → γ has y as its endpoint in ∂X ρ . It follows that y ∈ H B . Since H B ⊂ X ρ , we in fact have y ∈ ∂H B . Thus ∂H B = ¯ B . (cid:3) We remind the reader that in general ¯ B = B Z ( z, r ) may be a proper subset of the closedball ¯ B Z ( z, r ) = { x ∈ Z : d ( x, z ) ≤ r } . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 27
We next show that the closure of the hull of a ball B ⊂ Z can be approximated by balls in¯ X ρ centered at the center of B . We then use this to show that balls in ¯ X ρ centered at pointsof Z can be approximated by the closures of hulls of balls in Z . For a ball B = B Z ( z, r ) in Z we write ˆ B = B ρ ( z, r ) for the corresponding ball centered at z in ¯ X ρ . Lemma 6.13.
There is a constant C = C ( a, τ ) ≥ such that if B is any ball in Z then C − ˆ B ⊂ H B ⊂ C ˆ B . Consequently we have (6.3) H C − B ⊂ ˆ B ⊂ H CB . Proof.
Let B = B Z ( z, r ) be a given ball. Let 0 < λ < v ∈ V is a vertex satisfying d ρ ( v, z ) < λ r . Then by Lemma6.8 we have that a ( v | z ) h ≍ d ρ ( v, z ) < λ r. By inequality (2.9) it follows that(6.4) a h ( v ) . λ r, as well. We have by inequality (6.4) and Lemma 6.10, d ( π ( v ) , z ) ≤ d ρ ( π ( v ) , v ) + d ρ ( v, z ) . a h ( v ) + λ r . λ r. Thus d ( π ( v ) , z ) ≤ C λ r and a h ( v ) ≤ C λ r for a constant C = C ( a, τ ) ≥ a and τ . We set λ = C − /
2. It then follows that we have d ( π ( v ) , z ) < r and a h ( v ) ≤ r . This implies that π ( v ) ∈ B , which means that B ( v ) ∩ B = ∅ . Since a h ( v ) ≤ r it then follows that v ∈ H B .We conclude that all vertices v ∈ λ ˆ B satisfy v ∈ H B . Let 0 < λ < λ be another givenparameter. Let x ∈ λ ˆ B ∩ X ρ be any given point and let v be a vertex satisfying | xv | ≤ .Then by Lemma 6.8 and inequality (6.4) (for λ instead of λ ) we have d ρ ( v, x ) . a ( v | x ) h . a h ( v ) . λr, from which it follows that d ρ ( v, z ) ≤ d ρ ( v, x ) + d ρ ( x, z ) . λr. Thus there is a constant C = C ( a, τ ) such that d ρ ( v, z ) < Cλr for any vertex v such thatthere is a point x ∈ λ ˆ B with | xv | ≤ . We set λ = C − λ . Then it follows that v ∈ λ ˆ B and therefore v ∈ H B . By the definition of the hull we then conclude that x ∈ H B . Thus λ ˆ B ∩ X ρ ⊂ H B . Finally, since λ < ∂X ρ = Z , if x ∈ λ ˆ B ∩ ∂X ρ = λB then x ∈ B andtherefore x ∈ H B by Lemma 6.12. This proves the inclusion λ ˆ B ⊂ H B .Now let v ∈ H B be any vertex. Let y ∈ B ( v ) ∩ B be a point in this intersection. Then d ( π ( v ) , z ) ≤ d ( π ( v ) , y ) + d ( y, z ) < τ a h ( v ) + r . r, since a h ( v ) ≤ r . Then by Lemma 6.10, d ρ ( v, z ) ≤ d ρ ( v, π ( v )) + d ( π ( v ) , z ) . a h ( v ) + r . r. Thus there is a constant C = C ( a, τ ) ≥ d ρ ( v, z ) ≤ Cr . It follows that v ∈ C ˆ B .If x ∈ H B is an arbitrary point then we can find a vertex v ∈ H B such that | xv | ≤ .Then v ∈ C ˆ B by our previous calculations. By Lemma 6.8 we have d ρ ( x, v ) . a ( x | v ) h . a h ( v ) ≤ r. Thus d ρ ( x, z ) ≤ d ρ ( x, v ) + d ρ ( v, z ) . r. It follows that x ∈ C ˆ B as well, for a possibly larger constant C = C ( a, τ ). Finally since ∂H B = ¯ B ⊂ B by Lemma 6.12, we conclude that there is a constant C = C ( a, τ ) suchthat H B ⊂ C ˆ B . This completes the proof of the first assertion.It remains to obtain the chain of inclusions (6.3). Let B be a given ball in Z and let C = C ( a, τ ) ≥ CB gives that ˆ B ⊂ H CB and applying the conclusion of the first part to the ball C − B gives H C − B ⊂ ˆ B . This establishes the inclusions (6.3). (cid:3) Lifting doubling measures
We start with a complete doubling metric measure space (
Z, d, ν ), meaning that (
Z, d )is a metric space equipped with a doubling Borel measure ν satisfying 0 < ν ( B ) < ∞ forall balls B in Z . We will write C ν for the doubling constant of ν . It’s easy to see for adoubling metric measure space ( Z, d, ν ) that the underlying metric space (
Z, d ) is doublingwith constant D = D ( C ν ), see for instance [21, Chapter 4.1]. Thus by Proposition 6.2 anyhyperbolic filling X of Z with parameters 0 < a < τ > max { , (1 − a ) − } will havevertex degree bounded above by N = N ( a, τ, C ν ).We fix parameters 0 < a < τ > max { , (1 − a ) − } and let X be a hyperbolicfilling of Z with these parameters as constructed in the previous section. We carry overall concepts and notation from the previous section. In particular we let X ρ denote theconformal deformation of X with conformal factor ρ ( x ) = a h ( x ) and isometrically identify Z with ∂X ρ via a biLipschitz change of metric on Z . Throughout the first part of this section(until Lemma 7.3) all implied constants will depend only on a , τ , and the doubling constant C ν for ν .Our first task in this section will be to lift the doubling measure ν to a uniformly locallydoubling measure µ on X that supports a uniformly local 1-Poincar´e inequality. To thisend we adapt the construction in [3, Section 10]. As before we write V = S n ∈ Z V n for thevertices of X and write E for the set of edges of X . We let L denote the Borel measure on X given by Lebesgue measure on each edge of X , recalling that each edge of X has unit length.The measure L can equivalently be thought of as the 1-dimensional Hausdorff measure on X .We define a measure ˆ µ on V by setting for each v ∈ V ,(7.1) ˆ µ ( { v } ) = ν ( B ( v )) , where we recall that if v = ( z, n ) then B ( v ) = B Z ( z, τ a n ). To simplify notation we will writeˆ µ ( v ) := ˆ µ ( { v } ). Our first lemma shows that adjacent vertices have comparable ˆ µ -measure. Lemma 7.1.
Let v, w ∈ V satisfy v ∼ w . Then ˆ µ ( v ) ≍ ˆ µ ( w ) . Proof.
By symmetry it suffices to verify the upper bound ˆ µ ( v ) . ˆ µ ( w ). Since v ∼ w wemust have | h ( v ) − h ( w ) | ≤
1, which implies that h ( v ) ≥ h ( w ) −
1. Since B ( v ) ∩ B ( w ) = ∅ ,we must then have d ( v, w ) < τ a h ( w ) − . Thus if z ∈ B ( v ) then d ( z, w ) ≤ d ( v, w ) + d ( z, v ) < τ a h ( w ) − . Thus B ( v ) ⊂ B Z ( w, τ a h ( w ) − ). Writing 3 τ a h ( w ) − = 3 a − ( τ a h ( w ) ), the doubling conditionon ν implies that ν ( B ( w )) ≍ B Z ( w, τ a h ( w ) − ) ≥ ν ( B ( v )) . This implies that ˆ µ ( v ) . ˆ µ ( w ). (cid:3) XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 29
We next smear out ˆ µ to a measure µ on X by setting, for a Borel set A ⊂ X ,(7.2) µ ( A ) = X v ∈ V X w ∼ v (ˆ µ ( v ) + ˆ µ ( w )) L ( A ∩ vw ) . Here vw denotes the edge connecting v to w . By Lemma 7.1 and the fact that X has vertexdegree bounded by N = N ( a, τ, C ν ), we obtain the useful comparison for any vertex v ∈ V and any w ∼ v ,(7.3) ν ( B ( v )) = ˆ µ ( v ) ≍ µ ( vw ) . We can now apply [3, Theorem 10.2] to directly obtain uniformly local doubling of µ anda uniformly local 1-Poincar´e inequality on X . Proposition 7.2.
For each R > there is a constant C = C ( a, τ, C ν , R ) ≥ such thatfor all balls B = B X ( x, r ) in X with < r ≤ R and every integrable function u on B withupper gradient g on B we have (7.4) µ (2 B ) ≤ C µ ( B ) , and (7.5) − Z B | u − u B | dµ ≤ C r − Z B g dµ. We let β > µ β on X ρ by, for x ∈ X ,(7.6) dµ β ( x ) = a βh ( x ) dµ ( x ) . We extend µ β to a measure on ¯ X ρ by setting µ β ( ∂X ρ ) = 0. We observe that we can applythe results of Sections 4 and 5 in this setting, as if we set ε = − log a and let b be a Busemannfunction on X based at ω such that b . = h as in (6.2) then X ε will be biLipschitz to X ρ bythe identity map with constant a − , as noted before. If we define a measure ˇ µ β on X by d ˇ µ β ( x ) = e − βεb ( x ) dµ ( x ) = a βb ( x ) dµ ( x ) , then the results of Sections 4 and 5 can be applied directly to ˇ µ β . But since d ˇ µ β dµ β ≍ a − β , it immediately follows that these results can be applied to µ β as well. We note for applyingthese results that X is δ -hyperbolic with δ = δ ( a, τ ), that K = , that ε = − log a , and that M = M ( a, τ ) by [14, Theorem 1.9].Our next goal will be to show that µ β is ∂ -controlled in the sense of Definition 4.6 forany β >
0. We start by estimating the measure of the hull of a ball in Z and use this toestimate the measure of balls centered at the boundary. We recall for a ball B = B Z ( z, r ) in Z that we write ˆ B = B ρ ( z, r ) for the corresponding ball in ¯ X ρ . Throughout the rest of thissection all implied constants will depend only on a , τ , C ν , and β . By the estimate (7.3) andthe fact that h is 1-Lipschitz, we obtain for any vertex v ∈ V and any edge e with v ∈ e ,(7.7) µ β ( e ) ≍ a βh ( v ) ν ( B ( v )) . A half-edge e ∗ in X is a geodesic segment in X of length starting from a vertex v ∈ V .By applying the definition (7.2) and using Lemma 7.1 we similarly obtain for any half-edge e ∗ ⊂ X ,(7.8) µ β ( e ∗ ) ≍ a βh ( v ) ν ( B ( v )) . Since X has vertex degree bounded by N = N ( a, τ, C ν ), we obtain the useful estimate fora vertex v ∈ V ,(7.9) µ β (cid:18) ¯ B X (cid:18) v, (cid:19)(cid:19) = X v ∈ e ∗ µ β ( e ∗ ) ≍ a βh ( v ) ν ( B ( v )) , where the sum is taken over all half-edges e ∗ starting from v . Lemma 7.3.
Let B be any ball in Z . Then we have (7.10) µ β ( H B ) ≍ r β ν ( B ) , and (7.11) µ β ( ˆ B ) ≍ r β ν ( B ) . Proof.
We first obtain the comparison (7.10). Let B = B Z ( z, r ) be the given ball. Let m bethe minimal integer such that a m ≤ r , so that we have a m ≍ r . We then sum the estimate(7.9) over all vertices v ∈ H B , noting that each point x ∈ H B satisfies | xv | ≤ for at leastone vertex v ∈ H B and at most two vertices. We then obtain that(7.12) µ β ( H B ) ≍ ∞ X n = m a βn X v ∈ H n ( B ) ν ( B ( v )) . To estimate the inner sum on the right, note by Lemma 6.3 that there is a constant M = M ( a, τ, C ν ) such that for any x ∈ Z we have that x belongs to at most M balls B ( v ) for v ∈ V n . Furthermore each ball B ( v ) for v ∈ H B has radius at most 2 τ r by the definition of H B , so for each n ∈ Z we must have that B ⊂ [ v ∈ H n ( B ) B ( v ) ⊂ τ B. By the doubling property of ν and the bounded overlap of the balls B ( v ) associated tovertices v ∈ H n ( B ) we conclude that X v ∈ H n ( B ) ν ( B ( v )) ≍ ν ( B ) . Applying this comparison, summing the resulting geometric series in (7.12), and then using a m ≍ r gives µ β ( H B ) ≍ r β ν ( B ) , as desired. To obtain the corresponding result for ˆ B , we use Lemma 6.13 together with thefact that µ β ( ∂X ρ ) = 0 to obtain that µ β ( H C − B ) ≤ µ β ( ˆ B ) ≤ µ β ( H CB ) , with C = C ( a, τ ). By (7.10) and the doubling property for ν we have µ β ( H ( C − B )) ≍ r β ν ( B ) and µ β ( H ( CB )) ≍ r β ν ( B ), so the comparison for µ β ( ˆ B ) follows. (cid:3) We let κ = κ ( a, τ ) be the constant determined by Lemma 3.7. We recall the keyDefinition 4.6 for the measure µ β to be ∂ -controlled. Proposition 7.4.
The measure µ β on X ρ is ∂ -controlled for all β > , with constant C ∂ ( κ ) for < κ ≤ κ depending only on a , τ , C ν , β , and κ . Consequently the metric measurespace ( ¯ X ρ , d ρ , µ β ) is doubling and the metric measure spaces ( X ρ , d ρ , µ β ) and ( ¯ X ρ , d ρ , µ β ) each support a 1-Poincar´e inequality with dilation constant 1, with constants depending onlyon a , τ , C ν and β . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 31
Proof.
Let β > κ with 0 < κ ≤ κ there exists a constant C ∂ ( κ )depending only on a , τ , C ν , β , and κ such that for any z ∈ Z , r >
0, and x ∈ X we havethat whenever x ∈ X satisfies B ρ ( x, κr ) ⊂ B ρ ( z, r ) and d ρ ( x ) ≥ κr it follows that µ β ( B ρ ( z, r )) . C ∂ ( κ ) a βh ( x ) µ ( B X ( x, . Throughout this proof we will allow implied constants to depend on κ in addition to a , τ , C ν , and β . It then follows from the assumptions that d ρ ( x ) ≍ r . Viewing B ρ ( z, r ) as ˆ B forthe ball B = B Z ( z, r ) in Z , we conclude from Lemma 7.3 that it suffices to find a constant C ′ ∂ ( κ ) satisfying the same conditions such that(7.13) r β ν ( B ) . C ′ ∂ ( κ ) a βh ( x ) µ ( B X ( x, d ρ ( x ) ≍ r , Lemma 6.9 gives us that a βh ( x ) ≍ r β . It thus suffices to find a constant C ′′ ∂ ( κ ) satisfying these same conditions suchthat(7.14) ν ( B ) . C ′′ ∂ ( κ ) µ ( B X ( x, . For this final inequality we observe that, since the edges of X have unit length, for any x ∈ X the ball B X ( x,
1) must contain at least one half-edge e ∗ in X . Let v be the vertexon a half-edge e ∗ that is contained in B X ( x, µ of ν to X , we conclude that ν ( B ) ≍ µ ( e ∗ ) ≤ µ ( B X ( x, . Since a h ( x ) ≍ r and h ( x ) . = h ( v ), it then follows that a h ( v ) ≍ r . Let y = π ( v ) ∈ Z bethe point underlying v in Z . Then by the doubling property of ν we have that ν ( B ( v )) ≍ ν ( B Z ( y, r )). We have from Lemma 6.8 and Lemma 6.10 that d ρ ( x, y ) . d ρ ( x, v ) + d ρ ( v , y ) . a h ( v ) . r, and therefore d ρ ( y, z ) . r since x ∈ B ρ ( z, r ). There is thus a constant C = C ( a, τ, C ν , β, κ ) ≥ B Z ( y, r ) ⊂ B Z ( z, Cr ) and B Z ( z, r ) ⊂ B Z ( y, Cr ). This implies by the doublingproperty of ν that ν ( B Z ( y, r )) ≍ ν ( B Z ( z, r )), with comparison constant depending only on a , τ , C ν , β , and κ . We conclude that inequality (7.14) holds with C ′′ ∂ ( κ ) of the desired form.We thus obtain that µ β is ∂ -controlled for all β > C ∂ ( κ ) depending only on a , τ , C ν , β , and κ , as desired.We now combine Proposition 7.2 with R = 1 and Proposition 4.9 to conclude that themetric measure space ( ¯ X ρ , d ρ , µ β ) is doubling with doubling constant C µ β depending onlyon a , τ , C ν , β , and the values of the function κ → C ∂ ( κ ) at κ = κ / κ = κ /
12. Since κ = κ ( a, τ ) and the function κ → C ∂ ( κ ) depends only on a , τ , C ν , and β , we conclude thatthe doubling constant C µ β depends only on a , τ , C ν , and β . Lastly we apply Proposition5.4 together with Proposition 7.2 with R = 1 to conclude that the metric measure spaces( X ρ , d ρ , µ β ) and ( ¯ X ρ , d ρ , µ β ) each support a 1-Poincar´e inequality with dilation constant 1and constant C PI depending only on a , τ , C ν , and β . (cid:3) We note the following corollary of Lemma 7.3 and Proposition 7.4, which provides anestimate for the µ β -measure of balls centered at any point of ¯ X ρ . Closeness below is takenwith respect to the metric d ρ . Corollary 7.5.
Let x ∈ ¯ X ρ and r > be given. If r ≥ d ρ ( x ) then we let z ∈ Z be a pointclosest to x , while if r ≤ d ρ ( x ) then we let v ∈ V be a vertex of X ρ nearest to x . Then inthe case r ≥ d ρ ( x ) we have (7.15) µ β ( B ρ ( x, r )) ≍ r β ν ( B Z ( z, r )) , while in the case r ≤ d ρ ( x ) we have (7.16) µ β ( B ρ ( x, r )) ≍ rd ρ ( x ) β − ν ( B ( v )) . Proof.
We first consider the case r ≥ d ρ ( x ). In this case we have that d ρ ( x, z ) ≤ r andtherefore B ρ ( z, r ) ⊂ B ρ ( x, r ) ⊂ B ρ ( z, r ) . The desired estimate then follows from Lemma 7.3 and the fact that ν is doubling.We now consider the case r ≤ d ρ ( x ). We can then use Lemmas 3.6 and 6.9 to obtain forsome constants C ∗ = C ∗ ( a, τ ) and C = C ( a, τ ) that B ρ (cid:18) x, r (cid:19) ⊂ B X (cid:18) x, C ∗ rd ρ ( x ) (cid:19) ⊂ B X ( x, C ) . Let G be a minimal subgraph of X such that B X ( x, C ) ⊂ G . Since X has vertex degreebounded by N = N ( a, τ, C ν ), G has a number of edges M = M ( a, τ, C ν ) uniformly boundedin terms a , τ , and C ν . Furthermore by Lemma 7.1 the measure µ β restricted to G isuniformly comparable to the measure a βh ( v ) ˆ µ ( v ) L| G with comparison constants dependingonly on a , τ , C ν , and β , recalling that L denotes the measure on X that restricts to Lebesguemeasure on each edge of X . The ball B ρ (cid:0) x, r (cid:1) can be written as a union of at most Md ρ -geodesics starting from x . These geodesic segments have length comparable to ra − h ( v ) in X since h ( y ) . = h ( v ) for y ∈ G . Applying µ β to these segments gives the upper bound B ρ (cid:18) x, r (cid:19) . ra ( β − h ( v ) ˆ µ ( v ) . On the other hand the ball B ρ ( x, r ) must contain at least one d ρ -geodesic segment σ oflength r (with respect to d ρ ). Evaluating µ β on σ gives the lower bound, B ρ (cid:18) x, r (cid:19) & ra ( β − h ( v ) ˆ µ ( v ) . Applying Lemma 6.9 again, we obtain a ( β − h ( v ) ≍ a ( β − h ( x ) ≍ d ρ ( x ) β − . Combining thiswith the equality ˆ µ ( v ) = ν ( B ( v )) gives the second estimate (7.16). (cid:3) For any doubling measure ν on Z there is always some Q > z ∈ Z and 0 < r ′ ≤ r ,(7.17) ν ( B ( z, r ′ )) ν ( B ( z, r )) ≥ C − (cid:18) r ′ r (cid:19) Q , for some constant C low ≥
1. See [21, Lemma 8.1.13]. One may always take Q = log C ν ,but it is possible that (7.17) holds for smaller values of Q . We say that ν has relative lowervolume decay of order Q if inequality (7.17) holds for all z ∈ Z , all 0 < r ′ ≤ r , and someimplied constant. The exponent Q functions as a kind of dimension for ν , especially for thepurpose of embedding theorems [18, Section 5].In the next lemma we obtain an estimate on the relative lower volume decay exponentfor µ β in terms of the corresponding exponent for ν . Compare [3, Lemma 10.6]. XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 33
Lemma 7.6.
Suppose that ν has relative lower volume decay of order Q > . Then µ β has relative lower volume decay of order Q β = max { , Q + β } on ¯ X ρ with constant C ′ low depending only on a , τ , β , Q , and the constant C low in (7.17) .Proof. This is a direct consequence of the estimates of Corollary 7.5. Throughout this proofimplied constants are allowed to also depend on Q and the constant C low in (7.17). Let x ∈ ¯ X ρ be given. If r ≤ d ρ ( x ) then r ′ ≤ d ρ ( x ) as well and applying the estimate (7.16) forboth of them gives µ β ( B ρ ( x, r ′ )) µ β ( B ρ ( x, r )) ≍ r ′ r . Thus in this case (7.17) holds with exponent 1.Similarly if r ′ ≥ d ρ ( x ) then r ≥ d ρ ( x ) and we can apply the estimate (7.15) to both ofthem. This gives, for a nearest point z ∈ Z to x with respect to the metric d ρ , µ β ( B ρ ( x, r ′ )) µ β ( B ρ ( x, r )) ≍ (cid:18) r ′ r (cid:19) β ν ( B ( z, r ′ )) ν ( B ( z, r )) & (cid:18) r ′ r (cid:19) Q + β . Thus (7.17) holds with exponent Q + β .Finally we consider the case that r ′ ≤ d ρ ( x ) and r ≥ d ρ ( x ). We set r ′′ = d ρ ( x ) and write µ β ( B ρ ( x, r ′ )) µ β ( B ρ ( x, r )) = µ β ( B ρ ( x, r ′′ )) µ β ( B ρ ( x, r )) µ β ( B ρ ( x, r )) µ β ( B ρ ( x, r ′′ )) . The first case can be applied to the first ratio and the second case can be applied to thesecond ratio. This gives µ β ( B ρ ( x, r ′ )) µ β ( B ρ ( x, r )) & r ′ r ′′ (cid:18) r ′′ r (cid:19) Q + β . If Q + β ≥ r ′ ≤ r ′′ this implies that r ′ r ′′ (cid:18) r ′′ r (cid:19) Q + β ≥ (cid:18) r ′ r (cid:19) Q + β . Thus (7.17) holds with exponent Q + β in this subcase. If Q + β ≤ r ′′ ≤ r to obtain r ′ r ′′ (cid:18) r ′′ r (cid:19) Q + β ≥ r ′ r . Thus (7.17) holds with exponent 1 in this subcase. (cid:3)
Finally we remark that we always have µ β ( X ρ ) = ∞ , even when ν ( Z ) < ∞ . For each n ∈ Z we let E n denote the set of all edges in V which have at least one vertex in V n . Wenote that the definition of a metric measure space forces 0 < ν ( Z ) ≤ ∞ , since any ball B ⊂ Z must satisfy 0 < ν ( B ) < ∞ . Proposition 7.7.
We have µ β ( E n ) ≍ a βn ν ( Z ) for each n ∈ Z . Consequently µ β ( X ρ ) = ∞ .Proof. By the estimate (7.7) we have for any edge e ∈ E n that µ β ( e ) ≍ a βn ν ( B ( v )) for avertex v ∈ e ∩ V n . Since X has vertex degree bounded by N = N ( a, τ, C ν ), since each vertex v ∈ V n is attached to at least one edge, and since the balls B ( v ) for v ∈ V n cover Z andhave bounded overlap by Lemma 6.3, we conclude that X e ∈ E n µ β ( e ) ≍ X v ∈ V n a βn ν ( B ( v )) ≍ a βn ν ( Z ) . This proves the main estimate. The conclusion µ β ( X ρ ) = ∞ follows by letting n → −∞ . (cid:3) Function spaces and capacities
In this section we review some material from [3, Section 9] and make some adjustmentsto deal with the fact that the spaces we will be considering are unbounded.8.1.
Newtonian spaces.
For this first part we will be expanding on the discussion fromthe introduction; we refer to [21] for more details. We start with a metric measure space(
Y, d, µ ). For p ≥ u : Y → [ −∞ , ∞ ] we will use the notation k u k L p ( Y ) = (cid:0)R Y | u | p dµ (cid:1) /p for the L p norm of u , and write L p ( Y ) for the associated L p spaceon Y . We will also say that functions u ∈ L p ( Y ) are p -integrable . We write L p loc ( Y ) for thelocal L p space on Y , defined to be the set of all measurable functions u : Y → [ −∞ , ∞ ]such that u | B ∈ L p ( B ) for all balls B ⊂ Y , where we consider B as being equipped with therestricted measure µ | B . For a subset G ⊂ Y we will always write χ G for the characteristicfunction of G . Remark . In order to condense notation throughout the paper we will be omitting themeasure from the notation for the function spaces that we consider. Thus we write L p ( Y ) = L p ( Y, µ ), etc. We will always assume that p ≥ p = ∞ . We will always be using a fixed choice of measureon each metric space that we consider. Balls B ⊂ Y in a metric measure space ( Y, d, µ )will always be considered as metric measure spaces (
B, d, µ | B ) with the restriction of thedistance and measure on Y to B . Similarly we will sometimes write “a.e.” for “ µ -a.e.” whenthe measure µ is understood.We next define the p -modulus for p ≥
1. Let Γ be a family of curves in Y . A Borelfunction ρ : Y → [0 , ∞ ] is admissible for Γ if for each curve γ ∈ Γ we have R γ ρ ds ≥
1. The p -modulus of Γ is then defined asMod p (Γ) = inf ρ Z Y ρ p dµ, with the infimum taken over all admissible Borel functions ρ for Γ. We say that Γ is p -exceptional if Mod p (Γ) = 0. A property P is said to hold for p -a.e. curve if the collectionof curves for which P fails is p -exceptional. A subset G ⊂ Y is p -exceptional if the familyof all nonconstant curves meeting G is p -exceptional.We recall the definition (1.3) of upper gradients in the introduction. It is natural to relaxthis definition by allowing an exceptional set of curves on which the inequality (1.3) canpotentially fail. For a function u : Y → [ −∞ , ∞ ] and an exponent p ≥ g : Y → [0 , ∞ ] is a p -weak upper gradient for u if the upper gradient inequality(1.3) holds for p -a.e. curve in Y . The following standard lemma shows that p -weak uppergradients of u are not far from being true upper gradients of u from the perspective of the L p norm. Lemma 8.2. [21, Lemma 6.2.2]
Let u : Y → [ −∞ , ∞ ] be a function and suppose that g is a p -weak upper gradient of u . Then there is a a monotone decreasing sequence of uppergradients { g n } of u , with g n ≥ g for each n , such that k g n − g k L p ( Y ) → . In particular if u has a p -integrable p -weak upper gradient then it has a p -integrable uppergradient. We will be using Lemma 8.2 implicitly in many of the statements that follow.If u has a p -integrable upper gradient g then it has a minimal p -weak upper gradient g u that satisfies g u ≤ g a.e. for all p -integrable p -weak upper gradients g of u . This minimal p -weak upper gradient is unique up to a set of measure zero. The Newtonian space ˜ N ,p ( Y ) XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 35 is the space of all measurable functions u : Y → [ −∞ , ∞ ] such that k u k L p ( Y ) < ∞ andsuch that u has a p -weak upper gradient g ∈ L p ( Y ). This space comes equipped with theseminorm(8.1) k u k N ,p ( Y ) = k u k L p ( Y ) + k g u k L p ( Y ) , where g u is a minimal p -weak upper gradient for u . Lemma 8.2 shows that this definitionis equivalent to the previous definition (1.4) that we gave for the norm k u k N ,p ( Y ) . Weemphasize that functions in ˜ N ,p ( Y ) are required to be defined pointwise everywhere, inconstrast to what is typically required in the standard Sobolev space theory. wWe write N ,p ( Y ) = ˜ N ,p ( Y ) / ∼ be the quotient by the equivalence relation u ∼ v if k u − v k N ,p = 0.The space N ,p ( Y ) will also be referred to as the Newtonian space, and we will engage in thestandard practice from the theory of L p spaces of not distinguishing the notation between afunction u ∈ ˜ N ,p ( Y ) and its corresponding equivalence class [ u ] ∈ N ,p ( Y ). Equipped withthe norm (8.1) the space N ,p ( Y ) is a Banach space [28]. As with L p and Sobolev spaces,we also define a local version ˜ N ,p loc ( Y ) consisting of those functions u : Y → [ −∞ , ∞ ] suchthat for each ball B ⊂ Y we have u | B ∈ ˜ N ,p ( B ).The C Yp -capacity of a set G ⊂ Y is defined as(8.2) C Yp ( G ) = inf u k u k pN ,p ( Y ) , with the infimum taken over all functions u ∈ ˜ N ,p ( Y ) satisfying u ≥ G . A property P is said to hold quasieverywhere (q.e.) if the set G of points at which it fails has zero p -capacity, i.e., it satisfies C Yp ( G ) = 0. By [21, Proposition 7.2.8] a set G has zero p -capacityif and only if µ ( G ) = 0 and G is p -exceptional. Two functions u, v ∈ ˜ N ,p ( Y ) satisfy u ∼ v if and only if u = v quasieverywhere [21, Proposition 7.1.31]. Furthermore if u = v a.e. then u = v q.e. Thus the C Yp -capacity captures the degree of ambiguity one is allowed in thedefinition of Newtonian functions in N ,p ( Y ).The Dirichlet space ˜ D ,p ( Y ) consists of all measurable functions u : Y → [ −∞ , ∞ ] suchthat u has an upper gradient g ∈ L p ( Y ), or equivalently, such that u has a p -weak uppergradient g ∈ L p ( Y ). We equip ˜ D ,p ( Y ) with the seminorm k u k D ,p ( Y ) = k g u k p = inf g k g k p , with g u denoting a minimal p -weak upper gradient for u and the infimum being taken overall p -integrable upper gradients of u , or equivalently, over all p -integrable p -weak uppergradients of u . This matches our first definition (1.5) by Lemma 8.2. Similarly to N ,p ( Y ),we have for u, v ∈ ˜ D ,p ( Y ) that u = v a.e. if and only if u = v q.e. [21, Lemma 7.1.6].As with N ,p ( Y ), in order to obtain a norm we set D ,p ( Y ) = ˜ D ,p ( Y ) / ∼ , with u ∼ v if k u − v k D ,p ( Y ) = 0. We define a local version ˜ D ,p loc ( Y ) of the Dirichlet space exactly as we didfor the Newtonian space, writing u ∈ ˜ D ,p loc ( Y ) for a measurable function u : Y → [ −∞ , ∞ ]if u | B ∈ ˜ D ,p ( B ) for each ball B ⊂ Y . Remark . When Y is proper the spaces ˜ N ,p loc ( Y ) and ˜ D ,p loc ( Y ) can equivalently be de-scribed as the set of measurable functions u : Y → [ −∞ , ∞ ] such that each x ∈ Y has anopen neighborhood U x on which u | U x ∈ ˜ N ,p ( U x ) or u | U x ∈ ˜ D ,p ( U x ) respectively. This isthe definition of the local spaces given in [21, Chapter 7]. The definition of the local spacesthat we use here is more restrictive when Y is not proper. In the next section we will beapplying this definition in the particular case that Y is incomplete. In order to obtain further properties of these function spaces we need to make someadditional assumptions on Y . We will be assuming that Y is a geodesic metric spacethat contains at least two points, that µ is a doubling measure on Y , and that the metricmeasure space ( Y, d, µ ) supports a p -Poincar´e inequality for all measurable functions u : Y → [ −∞ , ∞ ] that are integrable on balls, all p -integrable upper gradients g : Y → [0 , ∞ ]for u , and all balls B ⊂ Y ,(8.3) − Z B | u − u B | dµ ≤ C PI diam( B ) (cid:18) − Z B g p dµ (cid:19) /p , for a constant C PI >
0. This corresponds to the previously considered p -Poincar´e inequality(5.1) with dilation constant λ = 1. If Y is geodesic then the stronger inequalityeqrefweak Poincare follows from the weaker inequality (5.1) [19, Theorem 4.18]. We notethat if ( Y, d, µ ) supports a p -Poincar´e inequality then ( Y, d, µ ) supports a q -Poincar´e in-equality for all q ≥ p by H¨older’s inequality, with new constants depending only on p , q ,and C PI .With these assumptions Y supports the following stronger form of the Poincar´e inequalityfor any ball B ⊂ Y , any integrable function u : B → R , and any p -integrable upper gradient g of u in B ,(8.4) (cid:18) − Z B | u − u B | p dµ (cid:19) /p ≤ C diam( B ) (cid:18) − Z B g p dµ (cid:19) /p , with the constant C > p -Poincar´e inequality for Y and the doubling constant for µ [21, Remark 9.1.19]. This follows by applying H¨older’sinequality to a class of Sobolev-Poincar´e inequalities for Y [21, Theorem 9.1.15] (see also[18]). This same application of H¨older’s inequality also shows that (8.3) similarly holdswhen u : B → [ −∞ , ∞ ] is defined only on B and g : B → [0 , ∞ ] is a p -integrable uppergradient of u on B . We will usually use the inequality (8.4) in the reformulated form,(8.5) Z B | u − u B | p dµ ≤ C p diam( B ) p Z B g p dµ. In keeping with standard conventions, we will refer to both (8.4) and (8.5) as ( p, p ) -Poincar´einequalities . We will use a generic constant C > C PI and C in (8.3) and (8.5) when applying these inequalities.The ( p, p )-Poincar´e inequality (8.5) implies by a straightforward truncation argumentthat all functions u ∈ ˜ D ,p loc ( Y ) are p -integrable over balls. The proof is very similar to theproof of [21, Lemma 8.1.5]. Proposition 8.4.
Let B ⊂ Y be any ball and let u : B → R be a measurable function suchthat u has an upper gradient g : B → [0 , ∞ ] with g ∈ L p ( B ) . Then u ∈ L p ( B ) . Consequently ˜ D ,p ( B ) = ˜ N ,p ( B ) (as sets). Therefore ˜ D ,p loc ( Y ) = ˜ N ,p loc ( Y ) and ˜ D ,p ( Y ) ⊂ ˜ N ,p loc ( Y ) .Proof. For each n ∈ N we let u n = max {− n, min { u, n }} . Then | u n | ≤ n and consequently u n is integrable over B . Furthermore g is also an upper gradient for u n on B [21, Proposition6.3.23]. Thus by the p -Poincar´e inequality (8.3) we have − Z B | u n − ( u n ) B | dµ ≤ C diam( B ) (cid:18) − Z B g p dµ (cid:19) /p . Since − Z B − Z B | u n ( x ) − u n ( y ) | dµ ( x ) dµ ( y ) ≤ − Z B | u n − ( u n ) B | dµ, XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 37 we can apply the monotone convergence theorem to the sequence of functions ϕ n : B × B → R given by ϕ n ( x, y ) = | u n ( x ) − u n ( y ) | to obtain that − Z B − Z B | u ( x ) − u ( y ) | dµ ( x ) dµ ( y ) ≤ C diam( B ) (cid:18) − Z B g p dµ (cid:19) /p < ∞ . It follows that the function x → | u ( x ) − u ( y ) | is integrable over B for a.e. y ∈ B , whichimmediately implies that u is integrable over B . Applying the ( p, p )-Poincar´e inequality(8.5) then gives u ∈ L p ( B ). We conclude that ˜ D ,p ( B ) = ˜ N ,p ( B ). The equality ˜ D ,p loc ( Y ) =˜ N ,p loc ( Y ) then follows from the definitions. The inclusion ˜ D ,p ( Y ) ⊂ ˜ N ,p loc ( Y ) then followsfrom the inclusion ˜ D ,p ( Y ) ⊂ ˜ D ,p loc ( Y ). (cid:3) The ( p, p )-Poincar´e inequality (8.5) allows us to show that D ,p ( Y ) is a Banach space.We will actually be able to show that D ,p ( B ) is a Banach space for any ball B ⊂ Y as well. Proposition 8.5.
The normed space D ,p ( Y ) is a Banach space. The same is true of D ,p ( B ) for any ball B ⊂ Y .Proof. We prove the second claim first. By Proposition 8.4 we have ˜ D ,p ( B ) = ˜ N ,p ( B )as sets. Since k u k D ,p ( B ) ≤ k u k N ,p ( B ) for any u ∈ ˜ D ,p ( B ), it follows that the quotientprojection N ,p ( B ) → D ,p ( B ) is continuous. Let { u n } be a Cauchy sequence in D ,p ( B ),and choose a sequence of representatives { ˜ u n } ⊂ ˜ D ,p ( B ) = ˜ N ,p ( B ). Since constantfunctions have norm 0 in ˜ D ,p ( B ), by adding an appropriate constant to ˜ u n for each n wecan assume that (˜ u n ) B = 0 for all n .For each m, n ∈ N we let g m,n be a minimal p -weak upper gradient of ˜ u m − ˜ u n in B .Then inequality (8.5) implies for m, n ∈ N that Z B | ˜ u m − ˜ u n | p dµ ≤ C diam( B ) p Z B g pm,n dµ. By hypothesis the right side converges to 0 as m, n → ∞ . Thus { ˜ u n } defines a Cauchysequence in L p ( B ), which implies that { ˜ u n } defines a Cauchy sequence in ˜ N ,p ( B ). Since N ,p ( B ) is a Banach space, it follows that there is some function ˜ u ∈ N ,p ( B ) such that˜ u n → ˜ u in N ,p ( B ). Letting u denote the projection of ˜ u to D ,p ( B ), this implies that u n → u in D ,p ( B ). It follows that D ,p ( B ) is a Banach space.Now let { u n } be a Cauchy sequence in D ,p ( Y ) and choose a sequence of representatives { ˜ u n } ⊂ ˜ D ,p ( Y ). Fix a point x ∈ Y and for each k ∈ N let B k = B ( x, k ) be the ball of radius k centered at x . By adding appropriate constants to ˜ u n we can arrange that (˜ u n ) B = 0for each n . For each n, k ∈ N we then set ˜ u n,k = ˜ u n − (˜ u n ) B k . Since { ˜ u n,k } also defines aCauchy sequence in D ,p ( B k ), the argument in the first part of the proof then shows thatfor each k ∈ N there is a function v k ∈ ˜ N ,p ( B k ) such that k (˜ u n,k − v k ) | B k k N ,p ( B k ) → n → ∞ .Restricting ˜ u n,k to B j for some j < k shows that k (˜ u n,k − v k ) | B j k N ,p ( B j ) = k (˜ u n − (˜ u n ) B k − v k ) | B j k N ,p ( B j ) → . Since k (˜ u n,j − v j ) | B j k N ,p ( B j ) → n →∞ k ((˜ u n ) B k − (˜ u n ) B j − v k + v j ) | B j k N ,p ( B j ) = 0 . Applying this to the special case j = 1, we obtain thatlim n →∞ k ((˜ u n ) B k − v k + v ) | B k N ,p ( B ) = 0 . It follows that for each k ∈ N there is a constant c k ∈ R such that v k − v = c k a.e. on B and c k = lim n →∞ (˜ u n ) B k . Note that c = 0. Applying this to (8.6), we conclude that v k − v j = c k − c j a.e. on B j . We define ˜ u : Y → R by setting ˜ u = v k − c k on B k ; sincefor j ≤ k we have v k − c k = v j − c j a.e. on B j ⊂ B k it follows that ˜ u is a well-definedmeasurable function on Y .It remains to show that ˜ u ∈ ˜ D ,p ( Y ) and that, denoting its projection to D ,p ( Y ) by u , we have u n → u in D ,p ( Y ). By construction we have that u n → u in D ,p ( B k ) foreach k ∈ N since u n | B k ∼ ˜ u n,k and ˜ u | B k ∼ v k in D ,p ( B k ) for each k . Letting g k denote aminimal p -weak upper gradient of ˜ u | B k in D ,p ( B k ), it follows that we will have k g k k L p ( B k ) ≤ lim sup n →∞ k u n k D ,p ( B k ) ≤ lim sup n →∞ k u n k D ,p ( Y ) < ∞ , with the final inequality following from the assumption that { u n } is a Cauchy sequence in D ,p ( Y ). By uniqueness of minimal p -weak upper gradients we have that g j = g k a.e. on B j for each j < k . We then extend g k to Y by setting g k ( x ) = 0 for x / ∈ B k and define a Borelfunction g : Y → [0 , ∞ ] by g ( x ) = sup k ∈ N g k ( x ). Then g defines a p -weak upper gradient for˜ u on Y . Furthermore we have g = g k a.e. on B k for each k . By the monotone convergencetheorem applied to the sequence of functions g ( k ) ( x ) = sup ≤ j ≤ k g j ( x ), we then concludethat k g k L p ( Y ) ≤ lim sup n →∞ k u n k D ,p ( Y ) < ∞ Thus ˜ u ∈ ˜ D ,p ( Y ).We thus conclude that ˜ u − ˜ u n ∈ ˜ D ,p ( Y ) for each n ∈ N . We let f n be a minimal p -integrable p -weak upper gradient of ˜ u − u n on Y for each n . To show that u n → u in D ,p ( Y ) it suffices to show that k f n k L p ( Y ) →
0. For each k ∈ N we have that f n | B k is aminimal p -weak upper gradient of (˜ u n,k − v k ) | B k . It follows that k f n k L p ( B k ) = k (˜ u n,k − v k ) | B k k D ,p ( B k ) ≤ lim sup m →∞ k (˜ u n,k − ˜ u m,k ) | B k k D ,p ( B k ) = lim sup m →∞ k (˜ u n − ˜ u m ) | B k k D ,p ( B k ) ≤ lim sup m →∞ k ˜ u n − ˜ u m k D ,p ( Y ) . By the monotone convergence theorem it follows that k f n k L p ( Y ) ≤ lim sup m →∞ k ˜ u n − ˜ u m k D ,p ( Y ) . The right side converges to 0 as n → ∞ since { u n } is a Cauchy sequence in D ,p ( Y ). Weconclude that k f n k L p ( Y ) → n → ∞ , which implies that u n → u in D ,p ( Y ). (cid:3) We next discuss quasicontinuity. A function u on Y is C Yp -quasicontinuous if for each η > U ⊂ Y such that C Yp ( U ) < η and u | Y \ U is continuous. We willmake use of the following theorem of Bj¨orn-Bj¨orn-Shanmugalingam. Theorem 8.6. [4]
Let ( Y, d, µ ) be a metric measure space that is complete, doubling, andsupports a p -Poincar´e inequality. Then every u ∈ ˜ N ,p ( Y ) is C Yp -quasicontinuous. Moreover C Yp is an outer capacity, i.e., for any subset G ⊂ Y , C Yp ( G ) = inf U C Yp ( U ) , with the infimum being taken over all open subsets U with G ⊂ U . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 39
The quasicontinuity statement of Theorem 8.6 can be extended to u ∈ ˜ D ,p ( Y ). For thiswe introduce an important construction that we will also use in the next section. For afunction ψ : Y → R defined on a metric space ( Y, d ) we writesupp( ψ ) := { y ∈ Y : ψ ( y ) = 0 } , for the support of ψ , defined as the closure of the points on which ψ is nonzero. We say that ψ has bounded support if supp( ψ ) is a bounded subset of Y , and we say that ψ is compactlysupported if supp( ψ ) is compact. These two notions coincide when Y is proper. Proposition 8.7. [27, Lemma B.7.4]
Let ( Y, d ) be a doubling metric space with doublingconstant D . Let r > be given. Let { y n } n ∈ J be a maximal r -separated subset of Y indexedby J ⊂ N . Then there is a corresponding collection { ψ n } n ∈ J of functions ψ n : Y → [0 , such that for each n ∈ J we have (8.7) supp( ψ n ) ⊂ B ( y n , r ) , and for each y ∈ Y , (8.8) X n ∈ J ψ n ( y ) = 1 , and ψ n is Cr − -Lipschitz for each n ∈ S with C = C ( D ) depending only on the doublingconstant N . The collection of functions { ψ n } n ∈ J will be referred to as a Lipschitz partition of unity .As noted after [10, (10)] the condition on supp( ψ n ) can be obtained by a slight modificationof the proof in the reference. We remark that by Lemma 6.3 the sum (8.8) always hasonly finitely many nonzero terms for each fixed choice of y ∈ Y . We also note that if Y isunbounded then we can always choose the index set J to satisfy J = N by renumbering theindices. Proposition 8.8.
Let ( Y, d, µ ) be a geodesic metric measure space that is complete, doubling,and supports a p -Poincar´e inequality. Then every u ∈ ˜ D ,p ( Y ) is C Yp -quasicontinuous.Proof. If Y is bounded then ˜ D ,p ( Y ) = ˜ N ,p ( Y ) by Proposition 8.4 and the claim thenfollows from Theorem 8.6. We can thus assume that Y is unbounded. Let u ∈ ˜ D ,p ( Y ) begiven and let g be a p -integrable upper gradient for u on Y . Let { ψ n } ∞ n =1 be a Lipschitzpartition of unity corresponding to a maximal 1-separated subset { y n } ∞ n =1 of Y . Let B n = B ( y n , u ∈ L p ( B n ) for each n ∈ N by Proposition 8.4.Let L be the upper bound for the Lipschitz constants of ψ n for n ∈ N given by Proposition8.7. We conclude that ψ n u ∈ ˜ D ,p ( B n ) for each n ∈ N since it is easy to see from theproduct rule for upper gradients [21, Proposition 6.3.28] that ψ n g + Lu defines a p -integrableupper gradient of ψ n u on B n . Applying Proposition 8.4 again, we conclude that ψ n u ∈ ˜ N ,p ( B n ) and therefore ψ n u ∈ ˜ N ,p ( Y ) since supp( ψ n ) ⊂ B n . Thus each function ψ n u is C Yp -quasicontinuous by Theorem 8.6.Now let η > n ∈ N we can find an open set U n ⊂ Y with C Yp ( Y \ U n ) < − n − η, such that ψ n u is continuous on Y \ U n . Setting U = S ∞ n =1 U n , we conclude that U is anopen subset of Y for which we have C Yp ( Y \ U ) < η since C Yp is an outer measure on Y [21, Lemma 7.2.4]. Thus each function ψ n u is continuous on Y \ U . By Lemma 6.3 we have ψ j = 0 on B n for all but finitely many indices j ∈ N . Since u = P n ∈ N ψ n u , it follows that u is continuous on B n \ U for each n and therefore u is continuous on Y \ U since each point x ∈ Y \ U has an open neighborhood U x with U x ⊂ B n \ U for some index n ∈ N . (cid:3) We will use the product rule for upper gradients frequently in the rest of the paper withoutfurther citation. We refer the reader to [21, Proposition 6.3.28] for the precise hypothesesunder which the product rule holds; for our purposes it suffices to note that the productrule holds for products of functions u and v that each have locally p -integrable p -weak uppergradients f and g respectively, in which case any Borel representative of g | u | + f | v | definesa p -weak upper gradient for the product uv (note that the function g | u | + f | v | need notitself be p -integrable). The reference requires that u and v are absolutely continuous along p -a.e. compact curve in Y , which follows immediately from the local p -integrability of f and g and [21, Proposition 6.3.2] since each compact curve in Y is contained in some ball B ⊂ Y on which f and g are p -integrable.8.2. Besov spaces.
In this section we will be dropping the requirement that our spacesupports a Poincar´e inequality, so we will be switching to different symbols Z for the metricspace and ν for the measure to reflect this. We will assume that ( Z, d, ν ) is a completedoubling metric measure space. As before we write C ν for the doubling constant of themeasure ν . For a given p ≥ θ > u ∈ L p loc ( Z ) the definitions(1.1) and (1.2) of the Besov norms k u k B θp ( Z ) and k u k ˇ B θp ( Z ) of u . The space ˜ B θp ( Z ) is thesubspace of L p loc ( Z ) for which we have k u k B θp ( Z ) < ∞ and the space ˇ B θp ( Z ) = L p ( Z ) ∩ ˜ B θp ( Z )is characterized by the finiteness of the norm k u k ˇ B θp ( Z ) .We let B θp ( Z ) = ˜ B θp ( Z ) / ∼ be the quotient of ˜ B θp ( Z ) by the equivalence relation u ∼ v if k u − v k B θp ( Z ) = 0. It is easy to see from the definition (1.1) that k u k B θp ( Z ) = 0 if and only ifthere is a constant c ∈ R such that u ≡ c a.e.; hence we can equivalently characterize B θp ( Z )as being the quotient of ˜ B θp ( Z ) / ∼ by the equivalence relation u ∼ v if u − v is constanta.e. on Z . It turns out that B θp ( Z ) is a Banach space; we will deduce this from Proposition8.5 in Proposition 11.1 at the end of the paper.We define the Besov capacity C Z ˇ B θp analogously to the C Yp -capacity for a subset G ⊂ ZC Z ˇ B θp ( G ) = inf u k u k p ˇ B θp ( Z ) , with the infimum being taken over all u ∈ ˇ B θp ( Z ) such that u ≥ ν -a.e. on a neighborhood of G . We have to take a neighborhood of G due to the lack of pointwise control over functions u ∈ ˇ B θp ( Z ). A function u ∈ ˜ B θp ( Z ) is C Z ˇ B θp -quasicontinuous if for each η > U ⊂ Z such that C Z ˇ B θp ( U ) < η and u | Z \ U is continuous.In connection with the Besov capacity, the following truncation lemma is useful. Lemma 8.9.
Let f ∈ ˜ B θp ( Z ) be given and define ˆ f = max { , min { , f }} . Then ˆ f ∈ ˜ B θp ( Z ) with k ˆ f k B θp ( Z ) ≤ k f k B θp ( Z ) and k ˆ f k ˇ B θp ( Z ) ≤ k f k ˇ B θp ( Z ) .Proof. For ν -a.e. x, y ∈ Z we have | ˆ f ( x ) − ˆ f ( y ) | ≤ | f ( x ) − f ( y ) | . This immediately implies that k ˆ f k B θp ( Z ) ≤ k f k B θp ( Z ) . Similarly we have | ˆ f ( x ) | ≤ | f ( x ) | for ν -a.e. x ∈ Z , which implies that k ˆ f k L p ( Z ) ≤ k f k L p ( Z ) and therefore k ˆ f k ˇ B θp ( Z ) ≤ k f k ˇ B θp ( Z ) . (cid:3) XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 41
We have the following key estimate, which follows from the estimates of [17, Theorem5.2] with a − > α = a − . Lemma 8.10.
For any < a < and u ∈ B θp ( Z ) we have k u k pB θp ( Z ) ≍ C X n ∈ Z Z Z − Z B Z ( x,a n ) | u ( x ) − u ( y ) | p a nθp dν ( y ) dν ( x ) , with C = C ( a, p, θ, C ν ) depending only on a , p , θ , and the doubling constant C ν of ν . We have the following useful consequence of Lemma 8.10, which shows for 0 < θ < Z with bounded support belong to ˇ B θp ( Z ). Lemma 8.11.
Suppose that < θ < . Let u : Z → R be an L -Lipschitz function for whichthere is some ball B = B Z ( z, R ) such that supp( u ) ⊂ B . Then (8.9) k u k B θp ( Z ) . C L ( R − θ + R θ ) ν ( B ) /p , with C = C ( p, θ, C ν ) . In particular u ∈ ˇ B θp ( Z ) .Proof. Let k ∈ Z be the greatest integer such that 2 − k ≥ R . Observe that we then have R ≍ − k . For m ≥ A m = { x ∈ Z : 2 m − k ≤ d ( x, z ) < m − k +1 } . We apply Lemma 8.10 with a = to obtain k u k pB θp ( Z ) ≍ C X n ∈ Z Z Z − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x )= X n ∈ Z Z B Z ( z, − k +1 ) − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x )+ ∞ X m =1 X n ∈ Z Z A m − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) , with C = C ( p, θ, C ν ). For x ∈ B Z ( z, − k +1 ) and n ≥ k − u is L -Lipschitz, − Z B ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) ≤ L p n ( θ − p . For n ≤ k − B Z ( z, R ) ⊂ B Z ( x, − k +2 ) since x ∈ B Z ( z, − k +1 ) and2 − k ≥ R . Thus u ( y ) = 0 for y / ∈ B Z ( x, − k +2 ). It follows that − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) = 1 ν ( B Z ( x, − n )) Z B Z ( x, − k +2 ) | u ( x ) − u ( y ) | p − nθp dν ( y ) ≤ ν ( B Z ( x, − k +2 )) ν ( B Z ( x, − n )) L p ( n − k +2) θp ≤ L p ( n − k +2) θp , since ν ( B Z ( x, − k +2 )) ≤ ν ( B Z ( x, − n )) because n ≤ k −
2. Thus, since 0 < θ < R ≍ − k , we have for x ∈ B Z ( z, − k +1 ), X n ∈ Z − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) ≤ ∞ X n = k − L p np ( θ − + k − X n = −∞ L p ( n − k +1) pθ . C L p (2 ( k − θ − p + 2 − ( k − θp ) . C L p ( R − ( θ − p + R θp ) . with C = C ( p, θ ) depending only on p and θ . It follows that(8.10) X n ∈ Z Z B Z ( z, − k +1 ) − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) . C L p ( R − ( θ − p + R θp ) ν ( B ) , with C = C ( p, θ, C ν ), using the doubling property of ν to obtain that ν ( B Z ( z, R )) ≍ C ν ν ( B Z ( z, − k +1 )) , since R ≍ − k .We now split into two cases. The first case is that in which Z = B Z ( z, R ). In this casewe have A m = ∅ for each m ≥ d ( x, z ) < R < − k +1 for all x ∈ Z . We thus derivefrom (8.10) that k u k pB θp ( Z ) . C L p ( R − ( θ − p + R θp ) ν ( B ) , with C = C ( p, θ, C ν ). Taking the p th root of each side and using the inequality ( s + t ) /p ≤ s /p + t /p for real numbers s, t ≥ Z = B Z ( z, R ). Let x ∈ A m for some m ≥
1. Then u ( x ) = 0.If y ∈ B Z ( x, − k + m − ) ∩ B Z ( z, − k ) then d ( x, z ) ≤ d ( x, y ) + d ( y, z ) < − k + m − + 2 − k ≤ − k + m − + 2 − k + m − = 2 − k + m , contradicting the definition of A m . Thus B Z ( x, − n ) ∩ B Z ( z, − k ) = ∅ whenever n ≥ k − m + 1. Since R ≤ − k , this implies that Z A m − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) = 0 , for n ≥ k − m + 1. Since u ( x ) = 0, for n < k − m + 1 we can use the upper bound Z A m − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) ≤ nθp k u k pL p ( Z ) ν ( A m ) ν ( B Z ( x, − n )) . Since for x ∈ A m we have A m ⊂ B Z ( z, m − k +1 ) ⊂ B Z ( x, m − k +2 ), it follows from thedoubling property of nu and the fact that n ≤ k − m that ν ( A m ) ν ( B Z ( x, − n )) ≤ ν ( B Z ( x, m − k +2 ) ν ( B Z ( x, − k + m )) ≤ C ν , and therefore Z A m − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) ≤ C ν nθp k u k pL p ( Z ) . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 43
Summing the geometric series over n ≤ k − m then gives X n ∈ Z Z A m − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) . C ( k − m ) θp k u k pL p ( Z ) , with C = C ( p, θ, C ν ). Summing this geometric series over m ≥ R ≍ − k then gives(8.11) ∞ X m =1 X n ∈ Z Z A m − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) . C R − θp k u k pL p ( Z ) , with C = C ( p, θ, C ν ).Since u is L -Lipschitz on Z and since u ( z ) = 0 for z ∈ Z \ B Z ( z, R ) (note we’ve assumedin this case that Z \ B Z ( z, R ) = ∅ ), we conclude that | u ( z ) | ≤ LR for z ∈ Z . It follows that k u k pL p ( Z ) ≤ p L p R p ν ( B ) . We thus conclude that(8.12) ∞ X m =1 X n ∈ Z Z A m − Z B Z ( x, − n ) | u ( x ) − u ( y ) | p − nθp dν ( y ) dν ( x ) . C L p R (1 − θ ) p ν ( B ) , with C = C ( p, θ, C ν ). Combining the estimates (8.10) and (8.12) together and then takingthe p th root of each side gives the desired estimate for k u k B θp ( Z ) . (cid:3) Trace theorems
In this section we carry over the concepts and notation from Section 7; we refer thereader back to the start of Section 7 for an overview of the setting and notation. Since weshowed in Proposition 7.4 that the geodesic doubling metric measure spaces ( X ρ , d ρ , µ β ) and( ¯ X ρ , d ρ , µ β ) each support a 1-Poincar´e inequality, they also support a p -Poincar´e inequalityfor each p ≥
1. Hence the results of the previous section apply to the metric measure spaces( X ρ , d ρ , µ β ) and ( ¯ X ρ , d ρ , µ β ).We will generalize the trace results of [3, Section 11] to the case of a potentially unboundedcomplete doubling metric space Z . Our treatment of the trace results in this context will beslightly different than that of [3], falling closer in spirit to the work of Bonk-Saksman [10]in that we will express the trace as a ν -a.e. limit of Lipschitz functions on Z . Throughoutthis section our parameter β corresponds to the ratio β/ε in [3, Section 11]. Remark . The statements from [3, Remark 9.5] carry through without modification toour setting, as they only rely on the fact that the metric measure space ( X ρ , d ρ , µ β ) is ametric graph with µ β being comparable to a multiple of Lebesgue measure on each edge.Given p ≥
1, the only family of nonconstant compact rectifiable curves in X ρ with zero p -modulus (with respect to µ β ) is the empty family. Thus any p -weak upper gradient for u on X ρ is an upper gradient for u . Since functions in N ,p loc ( X ρ ) are absolutely continuousalong p -a.e. curve [21, Proposition 6.3.2] this also implies that any function u ∈ ˜ N ,p loc ( X ρ )is continuous on X ρ and absolutely continuous along each edge of X ρ . Restricted to eachedge the minimal upper gradient g u of u is given by g u = (cid:12)(cid:12)(cid:12) duds ρ (cid:12)(cid:12)(cid:12) , with ds ρ denoting arclengthwith respect to the distance d ρ and duds ρ denoting the metric differential of u on this edgegiven by the absolute continuity of u on this edge (see [21, Theorem 4.4.8]). All points in X ρ have positive capacity, and each equivalence class in ˜ N ,p ( X ρ ) = N ,p ( X ρ ) consists of a single function that is continuous. Lastly all of these statements remain true of any metricsubgraph of X ρ , equipped with the restriction of the measure µ β to this subgraph.The following invaluable inequality is an immediate consequence of the convexity of thefunction t → t p for p ≥ , ∞ ). We will use it throughout this section largely withoutcomment. Lemma 9.2.
Let { x i } ki =1 be nonnegative real numbers. Then for any p ≥ we have k X i =1 x i ! p ≤ k p − k X i =1 x pi . Using Proposition 8.7 we fix, for each n ∈ Z , a Lipschitz partition of unity { ψ v } v ∈ V n associated to the a n -separated subset V n of Z , considering these vertices of X as points of Z . Since we require τ > ψ v ) ⊂ B ( v ) for all v ∈ V .Since 0 ≤ ψ v ≤ k ψ v k L ( Z ) ≤ ν ( B ( v )) . We make some additional definitions here for future reference. For n ∈ Z we write X ≥ n = X ∩ h − ([ n, ∞ )) for the set of all points in X of height at least n and write X ≤ n = X ∩ h − (( −∞ , n ]) for the set of all points in X of height at most n . We consider each ofthese subsets as being equipped with the metric d ρ . For a ball B ⊂ Z , its hull H B ⊂ X ρ ,and any integer n ∈ Z we then set H B ≥ n = H B ∩ X ≥ n and H B ≤ n = H B ∩ X ≤ n .All implied constants throughout this section will depend only on a , τ , C ν , β , and theexponent p ≥
1. We will always assume that p > β . By Remark 9.1 we have ˜ N ,p loc ( X ρ ) = N ,p loc ( X ρ ) and consequently ˜ N ,p ( X ρ ) = N ,p ( X ρ ).Our first proposition constructs the trace of a Newtonian function defined on the hull H B ⊂ X ρ of a ball B ⊂ Z . We recall from Lemma 6.12 that for a ball B ⊂ Z we have ∂H B = ¯ B . We will need the following lemma. Lemma 9.3.
Let e ∈ E be any edge of X and let v ∈ e denote either vertex on e . Thenrestricted to e we have the comparison (9.2) ds ρ | e ≍ a (1 − β ) h ( v ) dµ β | e ν ( B ( v )) , where ds ρ denotes arc length with respect to the metric d ρ .Proof. Let e ∈ E be a given edge and let v ∈ e denote a vertex on e . By the definition of X ρ we have on e ,(9.3) ds ρ | e ≍ a h ( v ) d L| e , where we recall that L is the measure on X given by Lebesgue measure on each unit lengthedge of X . On e we also have by the definition (7.2) of the measure µ and by Lemma 7.1,(9.4) d L| e ≍ dµ | e ˆ µ ( v ) = dµ | e ν ( B ( v )) ≍ dµ β | e a βh ( v ) ν ( B ( v )) . The comparison (9.2) follows by combining the comparisons (9.3) and (9.4). (cid:3)
For defining T n u in (9.5) below we recall that functions in Newtonian spaces are requiredto be defined pointwise everywhere; in this particular case the functions are actually con-tinuous by Remark 9.1. We consider H B as being equipped with the restriction µ β | H B ofthe measure µ β to this subset. XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 45
Proposition 9.4.
Let p > β be given. Let B ⊂ Z be any ball in Z of radius r > and let u ∈ N ,p ( H B ) be given. Then u has a trace T u ∈ L p ( B ) given as follows: for each n ∈ Z such that a n ≤ r and each z ∈ B we set (9.5) T n u ( z ) = X v ∈ V n u ( v ) ψ v ( z ) = X v ∈ H Bn u ( v ) ψ v ( z ) , then we have T n u → T u in L p ( B ) . Furthermore, letting k be the minimal integer such that a k ≤ r , we have the following estimate for any p -integrable upper gradient g of u on H B and any integer n ≥ k , (9.6) k T u − T n u k L p ( B ) . a (( p − β ) /p ) n k g k L p ( H B ≥ n ) . We remark that the second equality in (9.5) follows from the fact that ψ v ( z ) = 0 impliesthat z ∈ B ( v ) and therefore B ( v ) ∩ B = ∅ since z ∈ B , which implies that v ∈ H B providedthat a h ( v ) ≤ r . Proof.
We will first consider the case p = 1. Then 0 < β <
1. We will show in this case thatthe claim actually holds for β = 1 as well. Let u ∈ ˜ N , ( H B ) be a given function, whichis continuous by Remark 9.1. We let g ∈ L ( H B ) be an integrable upper gradient for u .For z ∈ B we write H Bn ( z ) for the set of v ∈ H Bn such that z ∈ B ( v ). We note that by thediscussion in the previous paragraph we have for all z ∈ B and n ∈ Z such that a n ≤ r , X v ∈ H Bn ( z ) ψ v ( z ) = 1 . We can then estimate(9.7) | T n +1 u ( z ) − T n u ( z ) | ≤ X v ′ ∈ H Bn +1 ( z ) X v ∈ H Bn ( z ) | u ( v ′ ) − u ( v ) | ψ v ′ ( z ) ψ v ( z ) . Observe that we can only have ψ v ′ ( z ) ψ v ( z ) = 0 if B ( v ) ∩ B ( v ′ ) = ∅ , which implies thatthere is a vertical edge e vv ′ from v to v ′ . Since g is an upper gradient for u on H B , we thusconclude in this case that(9.8) | u ( v ′ ) − u ( v ) | ≤ Z e vv ′ g ds ρ ≤ X e ∈U ( v ) Z e g ds ρ , with the sum being taken over the set U ( v ) of all upward directed vertical edges e startingfrom v , and with ds ρ denoting arclength with respect to the metric d ρ . By Lemma 9.3 thisimplies that(9.9) | u ( v ′ ) − u ( v ) | . a (1 − β ) n ν ( B ( v )) X e ∈U ( v ) Z e g dµ β . Applying the estimate (9.9) to the inequality (9.7) gives(9.10) | T n +1 u ( z ) − T n u ( z ) | . a (1 − β ) n X v ∈ H Bn ( z ) X e ∈U ( v ) Z e g dµ β ψ v ( z ) ν ( B ( v )) . Let m > n be a given integer. Summing inequality (9.10) above from n to m −
1, we obtainthat | T m u ( z ) − T n u ( z ) | . m − X j = n a (1 − β ) j X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g dµ β ψ v ( z )(9.11) ≤ ∞ X j = n a (1 − β ) j X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g dµ β ψ v ( z ) ν ( B ( v )) . (9.12)Integrating (9.11) over B and using the bound (9.1) then gives(9.13) k T m u − T n u k L ( B ) . ∞ X j = n a (1 − β ) j X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g dµ β . If β = 1 then it immediately follows that k T m u − T n u k L ( B ) . k g k L ( H B ≥ n ) . If 0 < β < k g k L ( H B ≥ n ) and sum theresulting geometric series to obtain(9.14) k T m u − T n u k L ( B ) . ∞ X j = n a (1 − β ) j k g k L ( H B ≥ n ) . a (1 − β ) n k g k L ( H B ≥ n ) . Since the sequence of sets { H ≥ n } ∞ n = k (for k the minimal integer such that a k ≤ r ) satisfies µ β ( H ≥ n ) → n → ∞ (since this is a nested sequence with empty intersection for whicheach set has finite measure) it follows that k g k L ( H B ≥ n ) → n → ∞ . Thus { T n u } definesa Cauchy sequence in L ( B ) which converges in L ( B ) to a function T u ∈ L ( B ). It followsthat for ν -a.e. z ∈ Z we have lim n →∞ T n u ( z ) = T u ( z ). Letting m → ∞ in the aboveinequalities, we deduce for any n ≥ k that k T u − T n u k L ( B ) . a (1 − β ) n k g k L ( H B ≥ n ) , This establishes (9.6) in the case p = 1. We conclude in particular that the propositionholds in the case p = 1.We now consider the case p >
1. We let u ∈ ˜ N , ( H B ) be a given function and let g ∈ L p ( H B ) be a p -integrable upper gradient for u . We start from the estimate (9.11) for m > n and z ∈ B . We let λ > p and q = pp − , we XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 47 obtain for m > n and z ∈ B , | T m u ( z ) − T n u ( z ) | . ∞ X j = n X v ∈ H Bj ( z ) a (1 − β − λ ) j X e ∈U ( v ) Z e g dµ β ψ v ( z ) ν ( B ( v )) a jλ ≤ ∞ X j = n a p (1 − β − λ ) j X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g dµ β ψ v ( z ) ν ( B ( v )) p /p ∞ X j = n a qjλ /q . a nλ ∞ X j = n a p (1 − β − λ ) j X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g dµ β ψ v ( z ) ν ( B ( v )) p /p , where now the implied constants also depend on λ . By Lemma 6.3 the sets H Bj ( z ) have anumber of elements uniformly bounded in a , τ , and C ν for each j ≥ n and z ∈ B . UsingLemma 9.2, it follows from this and the fact that X has vertex degree uniformly boundedin terms of this same data that we have X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g dµ β ψ v ( z ) ν ( B ( v )) p . X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g dµ β p ψ v ( z ) p ν ( B ( v )) p . X v ∈ H Bj ( z ) X e ∈U ( v ) (cid:18)Z e g dµ β (cid:19) p ψ v ( z ) ν ( B ( v )) p , where we have used 0 ≤ ψ v ( z ) ≤ ψ v ( z ) p ≤ ψ v ( z ). By Jensen’s inequality forintegrals of convex functions on probability spaces we have for any edge e ∈ X , (cid:18)Z e g dµ β (cid:19) p = µ β ( e ) p (cid:18) − Z e g dµ β (cid:19) p (9.15) ≤ µ β ( e ) p − Z e g p dµ β (9.16) = µ β ( e ) p − Z e g p dµ β (9.17)Using this estimate together with the comparison (7.7), we conclude that we have theinequality(9.18) | T m u ( z ) − T n u ( z ) | p . a λpn ∞ X j = n a ( p − β − pλ ) j X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g p dµ β ψ v ( z ) ν ( B ( v )) , again with the implied constant additionally depending on λ . For future reference we notethat inequality (9.18) also holds for p = 1 as it is implied by the inequality (9.11).We now set λ = ( p − β ) /p . This simplifies (9.18) to(9.19) | T m u ( z ) − T n u ( z ) | p . a ( p − β ) n ∞ X j = n X v ∈ H Bj ( z ) X e ∈U ( v ) Z e g p dµ β ψ v ( z ) ν ( B ( v )) . Integrating each side over B and using (9.1), we conclude that for all m > n ≥ k , k T m u − T n u k pL p ( B ) . a ( p − βn ∞ X j = n X v ∈ H Bj X e ∈U ( v ) Z e g p dµ β ≤ a ( p − β ) n k g k pL p ( H B ≥ n ) . By taking the p th root of each side, we obtain that(9.20) k T m u − T n u k L p ( B ) . a (( p − β ) /p ) n k g k L p ( H B ≥ n ) . In particular we have k T m u − T n u k L p ( B ) . a (( p − β ) /p ) n k g k L p ( H B ) . The right side converges to 0 as n → ∞ since p > β . We conclude that { T n u } defines aCauchy sequence in L p ( B ) which therefore converges in L p ( B ) to a function T u ∈ L p ( B ).Letting m → ∞ in (9.20) then gives (9.6). (cid:3) The proposition below is an immediate consequence of Proposition 9.4. We recall that˜ N ,p loc ( X ρ ) = ˜ D ,p loc ( X ρ ), so we will be formulating our theorems in terms of ˜ N ,p loc ( X ρ ). Proposition 9.5.
Let u ∈ ˜ N ,p loc ( X ρ ) be given with p > β . Then u has a trace T u ∈ L p loc ( Z ) given as follows: for each n ∈ Z and each z ∈ Z we set (9.21) T n u ( z ) = X v ∈ V n u ( v ) ψ v ( z ) . Then for each ball B ⊂ Z we have T n u → T u in L p ( B ) . Furthermore, for a given ball B ofradius r > and k the minimal integer such that a k ≤ r , we have the following estimatefor any p -integrable upper gradient g of u on H B and any integer n ≥ k , (9.22) k T u − T n u k L p ( B ) . a (( p − β ) /p ) n k g k L p ( H B ≥ n ) . If furthermore u has a p -integrable upper gradient g on X ρ then we have for each n ∈ Z , (9.23) k T u − T n u k L p ( Z ) . a (( p − β ) /p ) n k g k L p ( X ≥ n ) . Proof.
Let u ∈ ˜ N ,p loc ( X ρ ) be given. If B ⊂ Z is any ball of radius r > H B ⊂ C ˆ B for some C = C ( a, τ ) ≥
1, where ˆ B = B ρ ( z, r ) ⊂ ¯ X ρ , whichimplies for any x ∈ H B that H B ⊂ B ρ ( x, Cr ) ∩ X ρ . It follows that if u ∈ N ,p loc ( X ρ ) then u | H B ∈ N ,p ( H B ) for any ball B ⊂ Z . All of the claims of the proposition except forinequality (9.23) then follow immediately from the corresponding claims of Proposition 9.4upon observing that the formulas (9.5) and (9.21) defining T n u in each proposition are thesame.We now assume that u has a p -integrable upper gradient g on X ρ and let n ∈ Z be given.For each vertex v ∈ V n we let B ( v ) be the associated ball of radius r ( B ( v )) = τ a n , so thatin particular we have a n ≤ r ( B ( v )). Then g is a p -integrable upper gradient of u on H B ( v ) ,so inequality (9.22) implies that for each v ∈ V n we have k ( T u − T n u ) χ B ( v ) k L p ( Z ) . a (( p − β ) /p ) n k g k L p ( H B ( v ) ≥ n )XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 49 Since the balls B ( v ) for v ∈ V n cover Z with bounded overlap by Lemma 6.3, we deducefrom the triangle inequality in L p ( Z ) that k T u − T n u k L p ( Z ) . a (( p − β ) /p ) n X v ∈ V n k g k L p ( H B ( v ) ≥ n ) . To prove (9.23) it thus suffices to show that the subsets H B ( v ) ≥ n of X ≥ n for v ∈ V n also haveuniformly bounded overlap.Suppose that x ∈ H B ( v ) ≥ n ∩ H B ( w ) ≥ n for two vertices v, w ∈ V n . Since the midpoints ofall edges in X have µ β -measure zero, we can assume for the purpose of these estimatesthat x is not the midpoint of an edge in X . We then let v ′ be the closest vertex to x in X . We must then have v ′ ∈ H B ( v ) ≥ n ∩ H B ( w ) ≥ n as well, by the definition 6.11 of the hull.Thus B ( v ′ ) ∩ B ( v ) = ∅ and B ( v ′ ) ∩ B ( w ) = ∅ . Since r ( B ( v ′ )) ≤ r ( B ( v )), this implies that B ( w ) ⊂ B ( v ). Consequently π ( w ) ∈ B ( v ) for each w ∈ V n such that H B ( v ) ≥ n ∩ H B ( w ) ≥ n contains a point of X that is not the midpoint of an edge. The points π ( w ) then define an a n -separated subset of a ball 5 B ( v ) that has radius 5 τ a n . The desired uniform bound onthe number of such points π ( w ) then follows from the doubling property of ν (and thereforeof Z ). (cid:3) By Proposition 9.5 we have a linear trace operator T : ˜ N ,p loc ( X ρ ) → L p loc ( Z ) , defined by u → T u . The domain of T depends on both p and β , however we will suppressthis dependence in the notation.We next show that T restricts to a bounded linear operator T : N ,p ( X ρ ) → L p ( Z ); werecall from Remark 9.1 that each equivalence class in ˜ N ,p ( X ρ ) consists of a single continuousfunction, so we can consider N ,p ( X ρ ) to be canonically identified with ˜ N ,p ( X ρ ). Proposition 9.6.
Let u ∈ N ,p ( X ρ ) with p > β . Then T u ∈ L p ( Z ) with the estimate k T u k L p ( Z ) . k u k N ,p ( X ρ ) . Proof.
We define a function ξ : X ρ → [0 ,
1] by setting ξ ( x ) = 1 for x ∈ X ≥ , ξ ( x ) = 0 for x ∈ X ≤ , and linearly interpolating (with respect to the metric d ρ ) the values of ξ on eachvertical edge connecting X ≤ to X ≥ . On a given vertical edge e connecting v to w with h ( v ) = 0 and h ( w ) = 1 we have from Lemma 6.8 that d ρ ( v, w ) ≍
1. Thus there is a constant L = L ( a, τ ) such that ξ is L -Lipschitz on X ρ . Since ξ | X ≤ = 0, it follows that the scaledcharacteristic function Lχ X ≥ defines an upper gradient for ξ on X .Now set u ∗ = ξu . Then | u ∗ | ≤ | u | and therefore k u ∗ k L p ( X ρ ) ≤ k u k L p ( X ρ ) . Let g u be aminimal p -weak upper gradient for u on X ρ , which is an upper gradient for u on X ρ byRemark 9.1. The product rule for upper gradients implies that g ∗ := L | u | + g u ≥ Lχ X ≥ | u | + ξ · g u , is an upper gradient of u ∗ . It follows that k g ∗ k L p ( X ρ ) . k u k L p ( X ρ ) + k g u k L p ( X ρ ) = k u k N ,p ( X ρ ) . We conclude from the above that we have k u ∗ k N ,p ( X ρ ) . k u k N ,p ( X ρ ) .Since u ∗ | X ≥ = u | X ≥ , it follows immediately from the defining formula (9.21) for thetrace that we have T u ∗ = T u . On the other hand we have by construction that T u ∗ ≡ u ∗ ( v ) = 0 for all v ∈ V . It then follows from (9.23) applied in the case n = 0 that k T u k L p ( Z ) . k u ∗ k N ,p ( X ρ ) . k u k N ,p ( X ρ ) . (cid:3) Lipschitz functions on X ρ belong to ˜ N ,p loc ( X ρ ) and have a canonical extension by con-tinuity to ∂X ρ = Z . We show below that this extension agrees with the trace T . Theequality ˆ u | Z = T u below should be understood as holding for the distinguished L -Lipschitzrepresentative of T u in L p loc ( Z ). Proposition 9.7.
Let u : X ρ → R be L -Lipschitz and let ˆ u : ¯ X ρ → R denote the canonical L -Lipschitz extension of u to the completion ¯ X ρ of X ρ . Then ˆ u | Z = T u . In particular
T u is L -Lipschitz.Proof. We define T n u for each n ∈ Z as in (9.5). Let z ∈ Z be a given point and let γ z : R → X be an ascending geodesic line anchored at z as given by Lemma 6.4. Let { v n } n ∈ Z be the sequence of vertices on γ z with h ( v n ) = n . Then d ( π ( v n ) , z ) < a n for each n ∈ Z . For each n ∈ Z we then have by Lemma 6.10, d ρ ( v n , z ) ≤ d ρ ( v n , π ( v n )) + d ( π ( v n ) , z ) . a n . Since ˆ u is L -Lipschitz on ¯ X ρ it then follows that | u ( v n ) − ˆ u ( z ) | . La n . On the other hand we have | u ( v n ) − T n u ( z ) | ≤ X v ∈ V n | u ( v n ) − u ( v ) | ψ v ( z )The terms in the sum on the right are nonzero only when z ∈ B ( v ). Since z ∈ B ( v n ), thisimplies that z ∈ B ( v n ) ∩ B ( v ). Thus there is a horizontal edge e from v n to v . It thenfollows from Lemma 3.3 that d ρ ( v n , v ) . a n . We thus conclude in this case that | u ( v n ) − u ( v ) | ≤ Ld ρ ( v n , v ) . La n , which implies that | u ( v n ) − T n u ( z ) | . X v ∈ V n La n ψ v ( z ) = La n . Thus for all n ∈ Z and z ∈ Z we have | T n u ( z ) − ˆ u ( z ) | . La n . Since T n u → T u in L p ( ¯ B ), it follows in particular that T n u → T u pointwise a.e. on Z .We thus obtain that for any z ∈ Z such that lim n →∞ T n u ( z ) = T u ( z ) we in fact have T u ( z ) = ˆ u ( z ). Thus T u agrees ν -a.e. on Z with the L -Lipschitz function ˆ u | Z , as desired. (cid:3) We now estimate the Besov norm of the trace
T u for u ∈ ˜ D ,p ( X ρ ) ⊂ ˜ N ,p loc ( X ρ ). We notebelow that if p > β and we set θ = ( p − β ) /p then 0 < θ < Proposition 9.8.
Let u ∈ ˜ D ,p ( X ρ ) be given with p > β . Let θ = ( p − β ) /p . Then k T u k B θp ( Z ) . k u k D ,p ( X ρ ) . Consequently if u ∈ N ,p ( X ρ ) then T u ∈ ˇ B θp ( Z ) with the estimate k T u k ˇ B θp ( Z ) . k u k N ,p ( X ρ ) . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 51
Proof.
We start with u ∈ ˜ D ,p ( X ρ , µ β ) as specified. Since u ∈ ˜ D ,p loc ( X ρ ) = ˜ N ,p loc ( X ρ ) itfollows from Proposition 9.5 that the trace T u given by formula (9.21) exists and satisfies
T u ∈ L p loc ( Z ). We have the estimate for any n ∈ Z and ν -a.e. x, y ∈ Z , | T u ( x ) − T u ( y ) | p . | T u ( x ) − T n u ( x ) | p + | T n u ( x ) − T n u ( y ) | p + | T u ( y ) − T n u ( y ) | p . For n ∈ Z and x ∈ Z we define A n ( x ) = { y ∈ Z : a n +1 ≤ d ( x, y ) < a n } . We then have the following estimate for the Besov norm of
T u , using the doubling propertyof ν , k T u k pB θp ( Z ) . Z Z X n ∈ Z Z A n ( x ) | T u ( x ) − T n u ( x ) | p a nθp dν ( y ) dν ( x ) ν ( B ( x, a n ))+ Z Z X n ∈ Z Z A n ( x ) | T n u ( x ) − T n u ( y ) | p a nθp dν ( y ) dν ( x ) ν ( B ( x, a n ))+ Z Z X n ∈ Z Z A n ( y ) | T u ( y ) − T n u ( y ) | p a nθp dν ( x ) dν ( y ) ν ( B ( y, a n )) . Similarly to the proof of [3, Theorem 11.1], we label the three summands on the rightsequentially as ( I ), ( II ), and ( III ), and estimate each one separately. Since ( I ) and ( III )are related by switching the roles of x and y , it suffices to estimate ( I ) and ( II ).We begin with ( I ). Since none of the terms depend on y , we can integrate with respectto this variable and use the fact that ν ( A n ( x )) ≤ ν ( B ( x, a n )) (since A n ( x ) ⊂ B ( x, a n )) toobtain ( I ) . Z Z X n ∈ Z | T u ( x ) − T n u ( x ) | p a nθp dν ( x )We let g be a minimal p -weak upper gradient for u on X ρ , which is an upper gradient for u by Remark 9.1. We then apply inequality (9.18) to obtain for any choice of λ > I ) . Z Z X n ∈ Z a ( λ − θ ) pn ∞ X j = n a ( p − β − pλ ) j X v ∈ V j X e ∈U ( v ) Z e g p dµ β ψ v ( x ) ν ( B ( v )) dν ( x ) , with the implied constant additionally depending on λ . Using the bound (9.1), we concludefrom inequality (9.24) that( I ) . X n ∈ Z a ( λ − θ ) pn ∞ X j = n a ( p − β − pλ ) j X v ∈ V j X e ∈U ( v ) Z e g p dµ β . Using Tonelli’s theorem we can switch the order of summation to obtain( I ) . X j ∈ Z a ( p − β − pλ ) j X v ∈ V j X e ∈U ( v ) Z e g p dµ β j X n = −∞ a ( λ − θ ) pn . We set λ = θ/ p − β ) / p >
0. Summing the geometric series on the far right abovethen gives ( I ) . X j ∈ Z a ( p − β − pθ ) j X v ∈ V j X e ∈U ( v ) Z e g p dµ β . Since we assumed that θ = ( p − β ) /p , this simplies to the desired estimate ( I ) . k g k L p ( X ρ ) . We now estimate ( II ). For this we observe that | T n u ( x ) − T n u ( y ) | p . X v ∈ V n X v ′ ∈ V n | u ( v ) − u ( v ′ ) | p ψ pv ( x ) ψ pv ′ ( y ) ≤ X v ∈ V n X v ′ ∈ V n | u ( v ) − u ( v ′ ) | p ψ v ( x ) ψ v ′ ( y )using ψ v ≤
1, since this sum contains a number of terms uniformly bounded in terms of thenumber of v ∈ V n such that ψ v ( x ) = 0 and ψ v ( y ) = 0, which is uniformly bounded in termsof a , τ , and C ν by Lemma 6.3. Now suppose in addition that y ∈ B Z ( x, a n ), which followsfrom y ∈ A n ( x ). If ψ v ( x ) ψ v ′ ( y ) = 0 for some v, v ′ ∈ V n then we must have y ∈ B Z ( v ′ , a n )by the construction of the Lipschitz partition of unity in Proposition 8.7. Since d ( x, y ) < a n it then follows that x ∈ B Z ( v ′ , a n ) ⊂ B ( v ′ ) (since τ > x ∈ B ( v ),it then follows that B ( v ) ∩ B ( v ′ ) = ∅ . This implies that there is a horizontal edge in X from v to v ′ . Then, writing H ( v ) for the set of all horizontal edges having v as a vertex,using the comparison (9.2), using the Jensen inequality estimate (9.15), and then using thecomparison (7.7), we conclude that | T n u ( x ) − T n u ( y ) | p . X v ∈ V n X v ′ ∈ V n X e ∈H ( v ) (cid:18)Z e g ds α (cid:19) p ψ v ( x ) ψ v ′ ( z )= X v ∈ V n X e ∈H ( v ) (cid:18)Z e g ds α (cid:19) p ψ v ( x ) ≍ a p (1 − β ) n X v ∈ V n X e ∈H ( v ) (cid:18)Z e g dµ β (cid:19) p ψ v ( x ) ν ( B ( v )) p . a p (1 − β ) n X v ∈ V n X e ∈H ( v ) µ β ( e ) p − Z e g p dµ β ψ v ( x ) ν ( B ( v )) p ≍ a ( p − β ) n X v ∈ V n X e ∈H ( v ) Z e g p dµ β ψ v ( x ) ν ( B ( v )) . Using this estimate in ( II ) and using the fact that p − β = pθ , we conclude that( II ) . Z Z X n ∈ Z Z A n ( x ) X v ∈ V n X e ∈H ( v ) Z e g p dµ β ψ v ( x ) ν ( B ( v )) dν ( y ) dν ( x ) ν ( B ( x, a n )) , which, upon integrating with respect to y followed by x and using ν ( A n ( x )) ≤ ν ( B Z ( x, a n ))and the bound (9.1), gives( II ) . X v ∈ V n X e ∈H ( v ) Z e g p dµ β ≤ k g k L p ( X ρ ) . Since k g k L p ( X ρ ) = k u k D ,p ( X ρ ) , we conclude the desired estimate for k T u k B θp ( Z ) . The corre-sponding estimate for k T u k ˇ B θp ( Z ) then follows from Proposition 9.6. (cid:3) We conclude from Proposition 9.8 that the trace T : ˜ N ,p loc ( X ρ ) → L p loc ( Z ) restricts tobounded linear operators T : N ,p ( X ρ ) → ˇ B θp ( Z ) and T : D ,p ( X ρ ) → B θp ( Z ) when we set θ = ( p − β ) /p . In the next section we will show that these linear operators are surjective. XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 53
We can now generalize [3, Proposition 11.2] and [3, Theorem 11.3] to our setting. Thecompactness of Z and the uniformization ¯ X ρ are only used to show that the metric measurespaces ( X ρ , d ρ , µ β ) and ( ¯ X ρ , d ρ , µ β ) are doubling and support a 1-Poincar´e inequaliy, andto then derive the corresponding estimates of Proposition 9.8 in [3]. Once all of these claimshave also been verified in for the case of noncompact Z and ¯ X ρ , the proofs for [3, Proposition11.2] and [3, Theorem 11.3] carry over verbatim to the noncompact setting. Since the proofsare short, we reproduce abbreviated versions of them here for convenience for the reader.The first proposition shows that the boundary measure ν is absolutely continuous withrespect to the C ¯ X ρ p -capacity when p > β . We recall that our parameter β corresponds to β/ǫ in [3]. By Remark 9.1 any subset G ⊂ ¯ X ρ with C ¯ X ρ p ( G ) = 0 must satisfy G ⊂ Z . Proposition 9.9. [3, Proposition 11.2]
Let G ⊂ Z . If p > β and C ¯ X ρ p ( G ) = 0 then ν ( G ) = 0 .Proof. By Theorem 8.6 we can find open sets U n ⊂ ¯ X ρ for each n ∈ N with G ⊂ U n and C ¯ X p p ( U n ) < /n . The intersection G ′ = T ∞ n =1 U n then defines a Borel subset of ¯ X ρ with C ¯ X ρ p ( G ′ ) = 0, hence G ′ ⊂ Z . Let K ⊂ G ′ be compact. Since ( ¯ X ρ , d ρ , µ β ) supports a p -Poincar´e inequality, by [1, Theorem 6.7(xi)] we can find Lipschitz functions u k on ¯ X ρ suchthat u k = 1 on K and k u k k N ,p ( ¯ X ρ ) < /k . By Proposition 9.7 we have T u k = u k | Z , hencein particular T u k = 1 on K as well. Thus by Proposition 9.6 we have for each k , ν ( K ) /p ≤ k T u k k L p ( Z ) . k u k k N ,p ( ¯ X ρ ) < k . By letting k → ∞ we conclude that ν ( K ) = 0. Since G ′ is a Borel set and ν is a Borelregular measure on Z , we conclude that ν ( G ) ≤ ν ( G ′ ) = sup K ⊂ G ′ ν ( K ) = 0 , where the supremum is taken over all compact subsets K of G ′ . (cid:3) Since ( X ρ , d ρ , µ β ) is a uniform geodesic metric measure space that is doubling and satisfiesa p -Poincar´e inequality, by work of J. Bj¨orn and Shanmugalingam [6, Proposition 5.9] thereis a bounded linear operator assigning to any u ∈ N ,p ( X ρ ) an extension ˆ u ∈ ˜ N ,p ( ¯ X ρ ).We can then consider the restriction ˆ u | Z of this extension to Z . The next theorem givesa characterization of the trace T u ∈ ˇ B θp ( Z ) for u ∈ N ,p ( X ρ ) and shows that ˆ u | Z agrees ν -a.e. with T u . Proposition 9.10. [3, Theorem 11.3]
Let u ∈ N ,p ( X ρ ) with p > β and set θ = ( p − β ) /p . Then u has an extension ˆ u ∈ ˜ N ,p ( ¯ X ρ ) . This extension satisfies ˆ u | Z = T u ν -a.e. .Consequently we have the estimates k ˆ u | Z k B θp ( Z ) . k u k D ,p ( X ρ ) , and k ˆ u | Z k L p ( Z ) . k u k N ,p ( X ρ ) . Moreover, for C ¯ X ρ p -q.e. (and thus ν -a.e.) z ∈ Z we have (9.25) lim r → + − Z X ρ ∩ B ρ ( z,r ) | u − ˆ u ( z ) | p dµ β = 0 . The limit r → + indicates taking the limit through positive values of r . The extensionˆ u is unique up to sets of zero p -capacity in ¯ X ρ : if ˆ u and ˆ u are two such extensions thenˆ u = ˆ u µ β -a.e. on ¯ X ρ since µ β ( ∂X ρ ) = 0, which implies that they are equal q.e. on ¯ X ρ (see the discussion after (8.2)). This implies by Proposition 9.9 that the restriction ˆ u | Z isunique up to sets of ν -measure zero. In particular the restriction ˆ u | Z is well-defined as anelement of L p ( Z ).For a metric measure space ( Y, d, µ ) and an exponent p ≥ x ∈ Y isan L p ( Y ) -Lebesgue point of a measurable function u : Y → [ −∞ , ∞ ] iflim r → + − Z B Y ( x,r ) | u − u ( x ) | p dµ = 0 . If x is an L p ( Y )-Lebesgue point of u then by H¨older’s inequality it is also an L q ( Y )-Lebesguepoint of u for each 1 ≤ q ≤ p . Since µ β ( ∂X ρ ) = 0, (9.25) can be rephrased as saying that C ¯ X ρ p -q.e. point of Z is an L p ( ¯ X ρ )-Lebesgue point for ˆ u . Proof.
By [28, Theorem 4.1 and Corollary 3.9] we can find a sequence of Lipschitz functions u k ∈ ˜ N ,p ( ¯ X ρ ) such that k u k − ˆ u k N ,p ( ¯ X ρ ) → u k ( x ) → ˆ u ( x ) for q.e. x ∈ ¯ X ρ as k → ∞ .Let ˇ u k = u k | Z and ˇ u = ˆ u | Z . Then by Proposition 9.9 we have ˇ u k → ˇ u ν -a.e. on Z .By Proposition 9.7 we have T ( u j − u k ) = ˇ u j − ˇ u k for each j, k ∈ N . It follows fromProposition 9.6 that k ˇ u j − ˇ u k k ˇ B θp ( Z ) . k u j − u k k N ,p ( X ρ ) ≤ k u j − u k k N ,p ( ¯ X ρ ) Thus { ˇ u k } defines a Cauchy sequence in ˇ B θp ( Z ). Since ˇ B θp ( Z ) is a Banach space by [3,Remark 9.8], we conclude that this sequence converges in ˇ B θp ( Z ) to a function u ′ . In par-ticular we have ˇ u k → u ′ in L p ( Z ), which implies that ˇ u k → u ′ ν -a.e. on Z . Since we alsohave ˇ u k → ˇ u ν -a.e. on Z , we conclude that u ′ = ˇ u = ˆ u | Z ν -a.e. Since the trace T definesa bounded linear operator T : N ,p ( X ρ ) → ˇ B θp ( Z ) it follows that T u = ˆ u | Z in ˇ B θp ( Z ),i.e., T u = ˆ u | Z ν -a.e. The estimates for k ˆ u | Z k B θp ( Z ) and k ˆ u | Z k L p ( Z ) then follow from thecorresponding estimates for T u since
T u = ˆ u | Z ν -a.e.For p > C Yp -q.e. point of ¯ X ρ is an L p ( ¯ X ρ )-Lebesgue point of ˆ u [21, Theorem 9.2.8]. The same claim also holds for p = 1 [22, Theorem4.1 and Remark 4.7]; note in our case that µ β ( X ρ ) = ∞ by Proposition 7.7. This proves(9.25). (cid:3) Proposition 9.10 has a natural generalization to functions u ∈ ˜ D ,p ( X ρ ). Proposition 9.11.
Let u ∈ ˜ D ,p ( X ρ ) with p > β and set θ = ( p − β ) /p . Then u has anextension ˆ u ∈ ˜ D ,p ( ¯ X ρ ) . This extension satisfies ˆ u | Z = T u ν -a.e. . Consequently we havethe estimate (9.26) k ˆ u | Z k B θp ( Z ) . k u k D ,p ( X ρ ) , Moreover, for C ¯ X ρ p -q.e. (and thus ν -a.e.) z ∈ Z we have (9.27) lim r → + − Z X ρ ∩ B ρ ( z,r ) | u − ˆ u ( z ) | p dµ β = 0 . Proof.
Let u ∈ ˜ D ,p ( X ρ ) be given. We extend u to ˆ u : ¯ X ρ → [ −∞ , ∞ ] by setting for z ∈ Z ,ˆ u ( z ) = lim sup r → + − Z X ρ ∩ B ρ ( z,r ) u dµ β . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 55
The integrability of u over X ρ ∩ B ρ ( z, r ) follows from the fact that this is a bounded subsetof X ρ and that functions u ∈ ˜ D ,p ( X ρ ) are integrable on balls by Proposition 8.4.We fix z ∈ Z and for each n ∈ N let ζ n : [0 , ∞ ) → [0 ,
1] be a piecewise linear functionsuch that ζ n ( t ) = 1 for 0 ≤ t ≤ n , ζ n ( t ) = 2 − n − t for n ≤ t ≤ n , and ζ n ( t ) = 0 for t ≥ n .We then define κ n : ¯ X ρ → [0 ,
1] by κ n ( x ) = ζ n ( d ρ ( x, z )). We note that κ n is n − -Lipschitzfor each n . The rescaled characteristic function n − χ B ρ ( z , n ) defines an upper gradient for κ n on ¯ X ρ .For each n ∈ N we set f n = κ n u . The product rule for upper gradients implies that if g is any upper gradient for u on X ρ then for each n ∈ N ,(9.28) g n := n − χ B ρ ( z , n ) u + g, defines an upper gradient of f n on X ρ . We claim that if g is p -integrable then g n is p -integrable as well. For this it suffices to show that the function n − χ B ρ ( z , n ) u is p -integrableon X ρ for each n ∈ Z . But since for any point x ∈ B ρ ( z , n ) ∩ X ρ we have B ρ ( z , n ) ⊂ B ρ ( x, n ), this claim follows from the fact that functions in ˜ D ,p ( X ρ ) are p -integrable onballs by Proposition 8.4. Thus f n ∈ ˜ N ,p ( X ρ ) for each n ∈ N .We can therefore apply Proposition 9.10 to f n for each n ∈ N . We thus obtain anextension ˆ f n ∈ ˜ N ,p ( ¯ X ρ ) of f n for each n that satisfies for C ¯ X ρ p -q.e. (in particular ν -a.e.) z ∈ Z ,(9.29) lim r → + − Z X ρ ∩ B ρ ( z,r ) | f n − ˆ f n ( z ) | p dµ β = 0 . We let G ⊂ Z be the set of points such that (9.29) holds for each n ∈ N , which satisfies C ¯ X ρ p ( Z \ G ) by the countable subadditivity of the capacity. If z ∈ G then for sufficientlylarge n we will have from (9.29) and the fact that f n = u on B ρ ( z , n ),lim r → + − Z X ρ ∩ B ρ ( z,r ) | u − ˆ f n ( z ) | p dµ β = 0 . Applying H¨older’s inequality then giveslim r → + − Z X ρ ∩ B ρ ( z,r ) | u − ˆ f n ( z ) | dµ β = 0 , from which we conclude that ˆ f n ( z ) = ˆ u ( z ). We conclude in particular that the equality(9.27) holds. We also conclude that lim n →∞ ˆ f n ( z ) = ˆ u ( z ) for C ¯ X ρ p -q.e. z ∈ Z , hence also for ν -a.e. z ∈ Z .It is clear from the defining formula (9.22) for the trace T u that we have
T f n = T u ν -a.e. on B Z ( z , n/
2) since f n = u on B ρ ( z , n ). Thus we also have lim n →∞ T f n ( z ) = T u ( z )for ν -a.e. z ∈ Z . Since ˆ f n | Z = T f n ν -a.e. by Proposition 9.10, we conclude that ˆ u | Z = T uν -a.e. The estimate (9.26) immediately follows from this equality.Lastly we need to show that ˆ u ∈ ˜ D ,p ( ¯ X ρ ). Let g u be a minimal p -weak upper gradientof u on X ρ . For each n we let ¯ g n denote a minimal p -weak upper gradient of ˆ f n on ¯ X ρ .We then define a Borel function ¯ g : ¯ X ρ → [0 , ∞ ] by setting ¯ g ( x ) = sup k ≥ n ¯ g n ( x ) for x ∈ B ρ ( z , n/ \ B ρ ( z , ( n − /
2) and n ∈ N ; here by definition B ( z ,
0) = ∅ .We claim that ¯ g is a p -integrable p -weak upper gradient for ˆ u on ¯ X ρ . Let G ⊂ Z = ∂X ρ be the set we constructed earlier on which (9.29) holds for each n ∈ N . We then letˆ G = G ∪ X ρ . Since C ¯ X ρ p ( ¯ X ρ \ ˆ G ) = C ¯ X ρ p ( Z \ G ) = 0 we have that ¯ X ρ \ ˆ G is p -exceptional, i.e.,we have that p -a.e. curve in ¯ X ρ belongs entirely to ˆ G . Now let x, y ∈ ˆ G be given and let γ be a curve joining them in ˆ G . We can then choose n large enough that γ ⊂ B ρ ( z , n/ B ρ ( z , n/ ∩ ˆ G we have ˆ f n = ˆ u by the construction of ˆ f n and ˆ u . Thus since γ ⊂ B ρ ( z , n/ ∩ ˆ G and ¯ g n ≤ ¯ g on B ρ ( z , n/ | ˆ u ( x ) − ˆ u ( y ) | = | ˆ f n ( x ) − ˆ f n ( y ) | ≤ Z γ ¯ g n ds ≤ Z γ ¯ g ds. It follows that ¯ g is a p -weak upper gradient for ˆ u on ¯ X ρ .It remains to show that ¯ g is p -integrable on ¯ X ρ . Since µ β ( ∂X ρ ) = 0 it suffices to show that¯ g is p -integrable on X ρ . Since X ρ is open in ¯ X ρ it follows from [21, Proposition 6.3.22] thatfor each n ∈ N the minimal p -weak upper gradient ¯ g n for ˆ f n on ¯ X ρ coincides µ β -a.e. on X ρ with the minimal p -weak upper gradient ˇ g n of f n on X ρ . On the open set B ρ ( z , n/ ∩ X ρ we have that f n = u by construction. By using [21, Proposition 6.3.22] again it followsthat the minimal p -weak upper gradient ˇ g n for f n on X ρ coincides µ β -a.e. with the minimal p -weak upper gradient g u for u on X ρ when each of these upper gradients are restrictedto B ρ ( z , n/ ∩ X ρ . It follows that on B ( z , n/ ∩ X ρ we have ¯ g n = g u µ β -a.e. By thedefining formula for ¯ g we conclude that we have ¯ g = g u µ β -a.e. on X ρ . It follows that ¯ g is p -integrable on X ρ and therefore on ¯ X ρ . (cid:3) Extension theorems
In this section we establish analogues of the extension theorems in [3, Section 11] in thecase of noncompact Z . We carry over all conventions, notation, and hypotheses from theprevious section. Following Bonk-Saksman [10], for any function f ∈ L ( Z ) we define the Poisson extension
P f : X ρ → R by setting for any vertex v ∈ V , P f ( v ) = − Z B ( v ) f dν, and then extending P f to the edges of X ρ by linearly interpolating (with respect to arclengthfor the metric d ρ ) the values of P f on the vertices of each edge. Then
P f defines a continuousfunction on X ρ . We extend P f to ¯ X ρ by defining for z ∈ Z ,(10.1) P f ( z ) = lim sup r → + − Z B ρ ( z,r ) P f dµ β . Then
P f : ¯ X ρ → [ −∞ , ∞ ]. The resulting function defines a linear operator P : L ( Z ) → L ( ¯ X ρ ) since µ β ( ∂X ρ ) = 0. This operator is similar to the one used in [3, Section 11],however we use a different notation to avoid conflict with the notation E for the set of edgesin X .A computation similar to the one done in Proposition 7.7 shows that we can only have P f ∈ L p ( ¯ X ρ ) for some p ≥ f ≡ ν -a.e. on Z . We rectify this issue by defining truncations P n f of P f for each n ∈ Z as follows: we let ξ n : ¯ X ρ → [0 ,
1] be the functiondefined by setting ξ ( x ) = 1 for x ∈ X ≥ n +1 and x ∈ Z , ξ ( x ) = 0 for x ∈ X ≤ n , and linearlyinterpolating (with respect to the metric d ρ ) the values of ξ n on each vertical edge connecting X ≤ n to X ≥ n +1 . We then define P n f = ξ n P f . The operators f → P n f on L ( Z ) are alsolinear for each n ∈ Z . We have P n f | X ≤ n = 0 and P n f | X ≥ n +1 = P f | X ≥ n +1 as well as P n f | Z = P f | Z .In our first proposition of this section we analyze the L p norm of P f on the subsets X ≥ n for n ∈ Z . This estimate does not require p > β . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 57
Proposition 10.1.
Let f ∈ L p ( Z ) be given. Then for each n ∈ Z , k P f k L p ( X ≥ n ) . a ( β/p ) n k f k L p ( Z ) . Consequently, k P n f k L p ( X ρ ) . a ( β/p ) n k f k L p ( Z ) Proof. If v and w are two distinct vertices of the same edge e then, using Jensen’s inequalitytwice, Z e | P f | p dµ β ≤ µ β ( e )( | P f ( v ) | p + | P f ( w ) | p ) . µ β ( e ) − Z B ( v ) | f | p dν + − Z B ( w ) | f | p dν ! . Since v ∼ w , we have B ( v ) ∩ B ( w ) = ∅ and r ( B ( v )) ≍ a r ( B ( w )). Thus B ( w ) ⊂ a − B ( v ).It then follows from the doubling property of ν and the comparison (7.7) that Z e | P f | p dµ β . µ β ( e ) − Z a − B ( v ) | f | p dν . a βh ( v ) Z a − B ( v ) | f | p dν. As in Proposition 7.7 we let E j be the set of all edges in X with at least one vertex in V j .The balls 4 a − B ( v ) for v a vertex of some e ∈ E j then cover Z with bounded overlap byLemma 6.3. Using the doubling of ν again, it follows from summing over all such vertices v that for any j ∈ Z , Z E j | P f | p dµ β . a βj Z Z | f | p dν. Summing the geometric series over all integers j ≥ n and observing that each edge e ∈ X belongs to at most two of the sets E n for n ∈ Z then gives the first desired estimate. Theestimate for P n f follows by observing that | P n f | ≤ | P f | and P n f | X ≤ n = 0. (cid:3) We next show that
P f belongs to ˜ D ,p ( ¯ X ρ ) when f ∈ ˜ B θp ( Z ), where θ = ( p − β ) /p .We deduce from this that for f ∈ ˇ B θp ( Z ) the truncations P n f belong to ˜ N ,p ( ¯ X ρ ) for each n ∈ Z . Proposition 10.2. If f ∈ ˜ B θp ( Z ) with p > β and θ = ( p − β ) /p then (10.2) k P f k D ,p ( ¯ X ρ ) . k f k B θp ( Z ) . If f ∈ ˇ B θp,p ( Z ) then we further have for each n ∈ Z , (10.3) k P n f k D ,p ( ¯ X ρ ) . a − θn k f k L p ( Z ) + k f k B θp ( Z ) , and (10.4) k P n f k N ,p ( ¯ X ρ ) . ( a (1 − θ ) n + a − θn ) k f k L p ( Z ) + k f k B θp ( Z ) . Proof.
For each edge e of X with vertices v and w we set(10.5) g e = | P f ( v ) − P f ( w ) | d ρ ( v, w ) . The function g ( x ) = g e for x ∈ e belonging to the interior of e defines an upper gradient for Ef on the edge e by the construction of Ef . We define g : X ρ → R by setting g ( v ) = 0 foreach v ∈ V and setting g ( x ) = g e for each point x ∈ X belonging to the interior of an edge e . Since the vertices V of X have measure zero with respect to the 1-dimensional Hausdorff measure on X ρ , it follows that g defines an upper gradient for P f on X ρ . We note that g is constant on the interior of each edge of X .As noted in the proof of Proposition 10.1, if e is an edge of X with vertices v and w then B ( w ) ⊂ a − B ( v ). Since | vw | = 1, we also have from Lemma 6.8, d ρ ( v, w ) ≍ a ( v | w ) h ≍ a h ( v ) . Consequently by the doubling of ν , g e . a − h ( v ) − Z a − B ( v ) − Z a − B ( v ) | f ( x ) − f ( y ) | dν ( x ) dν ( y ) . By Jensen’s inequality we then have g pe . a − ph ( v ) − Z a − B ( v ) − Z a − B ( v ) | f ( x ) − f ( y ) | p dν ( x ) dν ( y )Multiplying and dividing by a pθh ( v ) and noting that pθ = p − β , we conclude that g pe . a − βh ( v ) − Z a − B ( v ) − Z a − B ( v ) | f ( x ) − f ( y ) | p a pθh ( v ) dν ( x ) dν ( y ) . Recall that L denotes the measure on X that restricts to 1-dimensional Lebesgue measureon each unit length edge of X . We have from the definition (7.2) of µ and Lemma 7.1 thatfor any edge e in X and either vertex v on e , dµ β | e ≍ a βh ( v ) ν ( B ( v )) d L| e . We thus have for any edge e in X and either vertex v on e , Z e g pe dµ β . Z a − B ( v ) − Z a − B ( v ) | f ( x ) − f ( y ) | p a pθh ( v ) dν ( y ) dν ( x ) . Now suppose that v ∈ V n . If x, y ∈ a − B ( v ) then y ∈ B Z ( x, τ a n − ). Thus Z e g pe dµ β . Z a − B ( v ) − Z B Z ( x, τa n − ) | f ( x ) − f ( y ) | p a pθn dν ( y ) dν ( x ) . Summing this inequality over the set E n of edges in X having at least one vertex in V n andusing the fact that the balls 4 a − B ( v ) for v a vertex of some e ∈ E n cover Z with boundedoverlap by Lemma 6.3, we obtain in a similar manner to what was done in Proposition 10.1, Z E n g pe dµ β . Z Z − Z B Z ( x, τa n − ) | f ( x ) − f ( y ) | p a pθn dν ( y ) dν ( x ) . By summing this inequality over n ∈ Z and using the fact that each edge e belongs to atmost two sets E n , we conclude that k g k pL p ( X ρ ) . X n ∈ Z Z Z − Z B Z ( x, τa n − ) | f ( x ) − f ( y ) | p a pθn dν ( y ) dν ( x ) . Let m = m ( a, τ ) ∈ N be the minimal integer such that a − m ≥ τ a − . Then by the doublingproperty for ν we have for each n ∈ Z , − Z B Z ( x, τa n − ) | f ( x ) − f ( y ) | p a pθn dν ( x ) . − Z B Z ( x,a n − m ) | f ( x ) − f ( y ) | p a pθ ( n − m ) dν ( x ) . It then follows by Lemma 8.10 that(10.6) k g k pL p ( X ρ ) . X n ∈ Z Z Z − Z B Z ( x,a n ) | f ( x ) − f ( y ) | p a pθn dν ( x ) dν ( y ) ≍ k f k pB θp ( Z ) . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 59
We conclude in particular that
P f ∈ ˜ D ,p ( X ρ ).Let u ∈ ˜ D ,p ( ¯ X ρ ) be an extension of P f to ¯ X ρ given by Proposition 9.11. Then, by (9.27)and the definition of P f on Z , we have that u = P f C ¯ X ρ p -q.e. on Z , from which it followsthat u = P f C ¯ X ρ p -q.e. on ¯ X ρ . Let g u be a minimal p -weak upper gradient for u on ¯ X ρ .Then g u must also be a p -weak upper gradient for P f on ¯ X ρ since p -a.e. curve in ¯ X ρ doesnot meet the p -exceptional set { u ( x ) = P f ( x ) } ⊂ ¯ X ρ . This shows that P f ∈ ˜ D ,p ( ¯ X ρ ).Now let g P f be a minimal p -weak upper gradient for P f on ¯ X ρ . Then g P f is also a p -integrable p -weak upper gradient of P f on X ρ . Since every compact curve in X ρ haspositive p -modulus (when considered as a family of curves with one element, see Remark9.1) it follows that g P f is actually a p -integrable upper gradient of P f on X ρ . Thus by(10.6) and the fact that µ β ( ∂X ρ ) = 0, k g P f k L p ( ¯ X ρ ) = k g P f k L p ( X ρ ) . k f k B θp ( Z ) . This proves the first estimate (10.2).Let n ∈ Z be given and consider the truncated extension P n f = ξ n P f . By Lemma 6.8 ifwe have vertices v ∈ X ≤ n and w ∈ X ≥ n +1 connected by a vertical edge then d ρ ( v, w ) ≍ a n .Thus ξ n is locally La − n -Lipschitz on ¯ X ρ for some constant L = L ( a, τ ) ≥
1, which impliesthat it is La − n -Lipschitz on ¯ X ρ since ¯ X ρ is geodesic. Since ξ n | X ≤ n = 0 and ξ | X ≥ n +1 ∪ Z = 1,we conclude that La − n χ X ≥ n ∪ Z defines an upper gradient for ξ n on ¯ X ρ . Let g P f be aminimal p -weak upper gradient for P f on ¯ X ρ . By the product rule for upper gradients wethen conclude from | ξ n | ≤ g n := La − n χ X ≥ n ∪ Z | P f | + g P f , is an upper gradient for P n f on ¯ X ρ . Since µ β ( Z ) = 0, we conclude from Proposition 10.1and the bound (10.2) that k g n k L p ( ¯ X ρ ) . a − n k P f k L p ( X ≥ n ) + k g P f k L p ( ¯ X ρ ) . a − n k P f k L p ( X ≥ n ) + k P f k D ,p ( ¯ X ρ ) . a − θn k f k L p ( Z ) + k f k B θp ( Z ) , since β/p − − θ . The bound (10.3) follows. The bound (10.4) is then a consequence ofProposition 10.1. (cid:3) We have thus defined a bounded linear operator P : ˜ B θp ( Z ) → D ,p ( ¯ X ρ ) as well asbounded linear operators P n : ˇ B θp ( Z ) → ˜ N ,p ( ¯ X ρ ) for each n ∈ Z . We will relate theseoperators to the trace operator T of the previous section using Proposition 10.3 below.This proposition shows, for each q ≥
1, that L q ( Z )-Lebesgue points for f ∈ L ( Z ) are L q ( ¯ X ρ )-Lebesgue points for P f . Proposition 10.3.
Let f ∈ L ( Z ) . Let z ∈ Z be an L q ( Z ) -Lebesgue point for f for agiven q ≥ . Then z is an L q ( ¯ X ρ ) -Lebesgue point for P and P f ( z ) = f ( z ) . Consequentlythe same is true with P n f replacing P f for each n ∈ Z .Proof. We closely follow the proof from the final part of [3, Theorem 12.1]. We let z ∈ Z bean L q ( Z )-Lebesgue point for f . We fix an arbitrary integer N ≥ x ∈ X such that d ρ ( x, z ) < a N . If x belongs to an edge e with vertices v and w then either v or w must belong to B ρ ( z, a N ) since the metric on ¯ X ρ is geodesic. Let v be the vertex belongingto B ρ ( z, a N ). Then(10.7) a h ( v ) ≍ d ρ ( v ) ≤ d ρ ( v, z ) < a N , so that we have a h ( v ) . a N . Then by Lemma 6.10 we have d ρ ( π ( v ) , z ) ≤ d ρ ( π ( v ) , v ) + d ρ ( v, z ) . a h ( v ) + a N . a N . A similar estimate shows that the other vertex w on e also satisfies d ( π ( w ) , z ) . a N . Since P f ( x ) is a convex combination of P f ( v ) and P f ( w ), we have by Jensen’s inequality, Z e | P f − f ( z ) | q dµ β ≤ ( | P f ( v ) − f ( z ) | q + | P f ( w ) − f ( z ) | q ) µ β ( e ) . Using Jensen again gives | P f ( v ) − f ( z ) | q ≤ − Z B ( v ) | f − f ( z ) | q dν, and the same with w replacing v . Combining this with the comparison (7.7) and using thefact that B ( w ) ⊂ a − B ( v ) gives Z e | P f − f ( z ) | q dµ β . a βn Z a − B ( v ) | f − f ( z ) | q dν. Summing this over all edges e such that at least one vertex of e belongs to B ρ ( z, a N ) and thenusing the fact that X has bounded degree and h ( v ) ≥ N − c for some constant c = c ( a, τ )(by (10.7)), we conclude that Z B ρ ( z,a N ) | Ef − f ( z ) | q dµ β . X n ≥ N − c X v ∈ V n ∩ B ρ ( z,a N ) a βn Z a − B ( v ) | f − f ( z ) | q dν. Let B = B Z ( z, a N ) and let C = C ( a, τ ) ≥ B ρ ( z, a N ) ⊂ H CB . Then for a given n ≥ N − c , the balls 4 a − B ( v ) for v ∈ H CBn havebounded overlap by Lemma 6.3 and will be contained in C ′ B for a constant C ′ = C ′ ( a, τ ).Thus Z B ρ ( z,a N ) | P f − f ( z ) | q dµ β . X n ≥ N − c a βn Z C ′ B | f − f ( z ) | q dν ≍ a βN Z C ′ B | f − f ( z ) | q dν. By combining Lemma 7.3 and the doubling property of ν we conclude that µ β ( B ρ ( z, a N )) ≍ a βN ν ( B Z ( z, a N )) ≍ a βN ν ( B Z ( z, C ′ a N )) . Thus − Z B ρ ( z,a N ) | P f − f ( z ) | q dµ β . − Z B Z ( z,C ′ a N ) | f − f ( z ) | q dν. Letting N → ∞ , we conclude that z is also an L q ( ¯ X ρ )-Lebesgue point for P f and that
P f ( z ) = f ( z ) (by the definition (10.1) of P f ( z )). The conclusions for P n f for each n ∈ Z then follow from the fact that P n f = P f on X ≥ n +1 ∪ Z , which implies that the L q ( ¯ X ρ )-Lebesgue points for P n f and P f on Z are the same. (cid:3) When specialized to Lipschitz functions, Propositions 10.2 and 10.3 show that the exten-sion
P f is also Lipschitz and restricts to f on Z . Proposition 10.4.
Let f : Z → R be L -Lipschitz. Then there is a constant C = C ( a, τ ) ≥ such that P f : ¯ X ρ → R is CL -Lipschitz. Furthermore we have P f | Z = f . XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 61
Proof.
For any vertex v ∈ V we have | P f ( v ) − f ( π ( v )) | ≤ − Z B ( v ) | f − f ( π ( v )) | dν ≤ Lτ a h ( v ) . Let e be an edge of X with vertices v and w . Then B ( v ) ∩ B ( w ) = ∅ , so we must have d ( π ( v ) , π ( w )) ≤ τ a h ( v ) − since | h ( v ) − h ( w ) | ≤
1. We then conclude that | P f ( v ) − P f ( w ) | ≤ | P f ( v ) − f ( π ( v )) | + | f ( π ( v )) − f ( π ( w )) | + | f ( π ( w )) − P f ( w ) |≤ τ a − La h ( v ) . We have d ρ ( v, w ) ≍ a h ( v ) by Lemma 6.8, with comparison constant depending only on a and τ . Thus if we define g e as in (10.5) then we conclude that g e ≤ CL with C = C ( a, τ ) ≥ P f is CL -Lipschitz in the metric d ρ on each edge of X ρ . Since X ρ isgeodesic it follows that P f is CL -Lipschitz on X ρ .Since f is Lipschitz we have that every point of Z is an L ( Z )-Lebesgue point for f .Proposition 10.3 then implies that every point of Z is an L ( ¯ X ρ )-Lebesgue point for P f .Fix a point z ∈ Z and let γ : [0 , ∞ ) → X be an ascending vertical geodesic ray anchoredat z as constructed in Lemma 6.4 with vertices v n = γ ( n ) on γ for each n ≥ h ( v n ) = n . Let r > n large enough that v n ∈ B ρ ( z, r ). Since P f is CL -Lipschitz on X ρ and since z ∈ B ( v n ), we have − Z B ρ ( z,r ) | P f − f ( z ) | dµ β ≤ CLr + | P f ( v n ) − f ( z ) |≤ CLr + − Z B ( v n ) | f − f ( z ) | dν ≤ CLr + CLτ a n . As r → n → ∞ , so we conclude thatlim r → + − Z B ρ ( z,r ) | P f − f ( z ) | dµ β = 0 . Since z is an L ( ¯ X ρ )-Lebesgue point for P f , it follows that
P f ( z ) = f ( z ). Since this holdsfor any z ∈ Z we conclude that P f | Z = f and that every point of ¯ X ρ is an L ( ¯ X ρ )-Lebesguepoint for P f .Let ˆ u denote the unique CL -Lipschitz extension of P f | X ρ to ¯ X ρ . Then every point of¯ X ρ is also an L ( ¯ X ρ )-Lebesgue point for ˆ u . Since µ β ( Z ) = 0, we have ˆ u = P f µ β -a.e. on¯ X ρ . Since every point of ¯ X ρ is an L ( ¯ X ρ )-Lebesgue point for both ˆ u and P f , we concludethat ˆ u = P f . This implies in particular that
P f is CL -Lipschitz on ¯ X ρ , as desired. (cid:3) Remark . For Lipschitz functions f : Z → R one can define a simpler Lipschitz extension˜ f : ¯ X ρ → R that does not make use of the measure ν by setting ˜ f ( v ) = f ( π ( v )) for eachvertex v ∈ V and linearly interpolating the values of ˜ f on the edges between vertices (withrespect to the metric d ρ ). This extension will have the same properties as the extension P f of f used in Proposition 10.4.We can now relate the trace and extension operators. Proposition 10.6.
Let f ∈ ˜ B θp ( Z ) for p > β and θ = ( p − β ) /p . Then T ( P f ) = f ν -a.e.Consequently the induced trace operators T : D ,p ( X ρ ) → B θp ( Z ) and T : N ,p ( X ρ ) → ˇ B θp ( Z ) are surjective. Proof.
Let f ∈ ˜ B θp ( Z ) be given. Since f ∈ L ( Z ) we have that ν -a.e. point of Z is an L ( Z )-Lebesgue point of f by the Lebesgue differentiation theorem [19, Theorem 1.8]. ByProposition 10.2 we have P f ∈ ˜ D ,p ( ¯ X ρ ), and by Proposition 10.3 we have that each L ( Z )-Lebesgue point z ∈ Z for f is an L ( ¯ X ρ )-Lebesgue point for P f that satisfies
P f ( z ) = f ( z ).We write u = P f | X ρ and let ˆ u denote the extension of u to ¯ X ρ given by Proposition 9.11.Then ˆ u | Z = T u ν -a.e. On the other hand we have by (9.27) and H¨older’s inequality thateach L ( ¯ X ρ )-Lebesgue point of P f on Z (and therefore each L ( Z )-Lebesgue point of Z )satisfies P f ( z ) = ˆ u ( z ). We conclude that P f | Z = ˆ u | Z = T ( P f ) ν -a.e. Since P f | Z = fν -a.e. , we conclude that T ( P f ) = f ν -a.e., as desired.Let f ∈ ˜ B θp ( Z ) be arbitrary. Then u = P f | X ρ defines an element of ˜ D ,p ( X ρ ) such that T u = f ν -a.e. on Z and therefore T u = f in B pθ ( Z ). We conclude that the trace operator T : D ,p ( X ρ ) → B θp ( Z ) is surjective.Now let f ∈ ˇ B θp ( Z ) be arbitrary. We set u = P f | X ρ . Then since u | X ≥ = P f | X ≥ ,we conclude that u defines an element of N ,p ( X ρ ) such that T u = f ν -a.e. on Z , whichimplies that T u = f in ˇ B θp ( Z ). Thus the trace operator T : N ,p ( X ρ ) → ˇ B θp ( Z ) is alsosurjective. (cid:3) Proposition 10.6 completes the proof of Theorem 10.6. The representative
P f | Z of f in ˜ B θp ( Z ) constructed in the proof of Proposition 10.6 is better behaved than the originalfunction f in many respects. We elaborate on this in the next section.As a first application of Proposition 10.6, we show that the Besov capacity can be com-puted in terms of the C ¯ X ρ p -capacity and vice versa. We recall that the implied constantsbelow depend only on a , τ , C ν , p , and β . Proposition 10.7.
Let p > β and set θ = ( p − β ) /p . Then for any set G ⊂ Z we have C ¯ X ρ p ( G ) ≍ C Z ˇ B θp ( G ) .Proof. Let G ⊂ Z be a given subset. Let f ∈ ˇ B θp ( Z ) be given such that f ≥ ν -a.e. on aneighborhood U of G . By truncating f using Lemma 8.9 and then redefining f on a ν -nullset, we can assume that we in fact have f ≡ U . Let P f ∈ ˜ N ,p ( ¯ X ρ ) be the extensionof f given by Proposition 10.2. Since every point of U is an L ( Z )-Lebesgue point for f ,we conclude by Proposition 10.3 that P f | U = f . Thus P f = 1 on U . It follows that P f isadmissible for the C ¯ X ρ p -capacity of G and therefore C ¯ X ρ p ( G ) ≤ k P f k pN ,p ( ¯ X ρ ) . k f k p ˇ B θp ( Z ) . Minimizing over all admissible f for the C Z ˇ B θp -capacity of G then gives C ¯ X ρ p ( G ) . C Z ˇ B θp ( G ).For the other direction, let ε > U ⊂ ¯ X ρ be an open set containing G such that C ¯ X ρ p ( U \ G ) < ε , which we can find by Theorem 8.6. We let u ∈ ˜ N ,p ( ¯ X ρ ) bea function such that u ≥ U and k u k N ,p ( ¯ X ρ ) < C ¯ X ρ p ( U ) + ε . Then u | Z ∈ ˇ B θp ( Z ) byProposition 9.10 and u ≥ U ∩ Z ⊂ Z of Z that contains G . Thus C Z ˇ B θp ( G ) ≤ k u | Z k ˇ B θp ( Z ) . k u k N ,p ( ¯ X ρ ) < C ¯ X ρ p ( U ) + ε< C ¯ X ρ p ( G ) + 2 ε. XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 63
Letting ε → (cid:3) Properties of Besov spaces
In this final section we apply the results of Sections 9 and 10 to establish a number ofproperties of the Besov spaces ˜ B θp ( Z ) on a complete doubling metric measure space ( Z, d, ν )for p ≥ < θ <
1. Most of these properties were previously established by Bj¨orn-Bj¨orn-Shanmugalingam in the case that Z is compact [3, Section 13]. In particular we proveCorollaries 1.2, 1.3, and 1.4.For this section we consider a complete doubling metric measure space ( Z, d, ν ). We let X be a hyperbolic filling of Z with parameters a = and τ = 4; these parameters satisfythe requirement (6.1), as can be easily checked. We let µ be the lift of the measure ν to X defined in (7.2). We then let X ρ be the uniformized hyperbolic filling corresponding to theseparameters as defined in Section 6. For a given p ≥ < θ < β = p (1 − θ ),noting that we then have p > β and θ = ( p − β ) /p . Note that in contrast to Sections 9 and10, we are considering β here as depending on p and θ instead of considering θ as dependingon p and β . We then let µ β be the measure on ¯ X ρ defined by (7.6).Throughout the rest of this section all implied constants will depend only on the doublingconstant C ν for ν as well as the exponents p ≥ < θ <
1. The uniformized filling X ρ will always be considered to be equipped with the measure µ β for β = p (1 − θ ), where p and θ are given in the hypotheses of each proposition.Our first application establishes that the homogeneous Besov space B θp ( Z ) = ˜ B θp ( Z ) / ∼ is a Banach space. Proposition 11.1. B θp ( Z ) is a Banach space for p ≥ and < θ < .Proof. Let { f n } be a Cauchy sequence in B θp ( Z ), and let { ˜ f n } be a sequence of representa-tives of these functions in ˜ B θp ( Z ). The sequence of functions { P ˜ f n } in ˜ D ,p ( ¯ X ρ ) then definesa Cauchy sequence in D ,p ( ¯ X ρ ) by Proposition 10.2 and the linearity of the extension op-erator P . Since D ,p ( ¯ X ρ ) is a Banach space by Proposition 8.5, we conclude that there is afunction u ∈ ˜ D ,p ( ¯ X ρ ) such that k P ˜ f n − u k D ,p ( ¯ X ρ ) →
0. Then u has a trace T u ∈ ˜ B θp ( Z )by Proposition 9.8. By Proposition 9.8 we also conclude that k T ( P ˜ f n ) − T u k B θp ( Z ) = k T ( P ˜ f n − u ) k B θp ( Z ) → . Since T ( P ˜ f n ) = ˜ f n ν -a.e. on Z by Proposition 10.6, this implies that k ˜ f n − T u k B θp ( Z ) → f denote the projection of T u to B θp ( Z ), we conclude that f n → f in B θp ( Z ). Itfollows that B θp ( Z ) is a Banach space. (cid:3) Our next application concerns the density of Lipschitz functions in B θp ( Z ). We note that Z is proper since it is complete and doubling, which implies that closed and bounded subsetsof Z are compact. Proposition 11.2.
Lipschitz functions with compact support are dense in ˇ B θp ( Z ) for p ≥ and < θ < .Proof. We first remark that Lipschitz functions with compact support belong to ˇ B θp ( Z ) byProposition 8.11. We next note that functions with compact support are dense in N ,p ( ¯ X ρ )by [21, Proposition 7.1.35] since the metric measure space ( ¯ X ρ , d ρ , µ β ) is complete, doubling,and supports a p -Poincar´e inequality. If u ∈ N ,p ( ¯ X ρ ) has compact support then the proofof [21, Theorem 8.2.1] shows that one can find a sequence of Lipschitz functions { u n } on ¯ X ρ with compact support such that u n → u in N ,p ( ¯ X ρ ). It follows that Lipschitz functionswith compact support are dense in N ,p ( ¯ X ρ ).Let f ∈ ˇ B θp ( Z ) be given. Let P f ∈ ˜ N ,p ( ¯ X ρ ) be the extension of f constructed inProposition 10.2. We can then find a sequence of Lipschitz functions { u n } on ¯ X ρ withcompact support such that k u n − P f k N ,p ( ¯ X ρ ) →
0. By Proposition 9.8 we then have k T u n − T ( P f ) k ˇ B θp ( Z ) →
0. By Proposition 9.7 we have
T u n = u n | Z for each n and byProposition 10.6 we have T ( P f ) = f ν -a.e. It follows that k u n | Z − f k ˇ B θp ( Z ) →
0. Sinceeach of the restrictions u n | Z are Lipschitz functions on Z with compact support, we concludethat Lipschitz functions with compact support are dense in ˇ B θp ( Z ). (cid:3) As we remarked after the proof of Proposition 10.6, given f ∈ ˇ B θp ( Z ) it is possible to finda representative of f with better regularity properties. Proposition 11.3 below makes thisprecise. Proposition 11.3.
For each f ∈ ˜ B θp ( Z ) there is a C Z ˇ B θp -quasicontinuous function ˇ f suchthat f = ˇ f ν -a.e. in Z .Proof. We let ˇ f = P f | Z be the restriction of P f to Z that was considered in the proof ofProposition 10.6; as shown in the proof we have ˇ f = f ν -a.e. on Z . Since P f ∈ ˜ D ,p ( ¯ X ρ ),we have by Proposition 8.8 that P f is C ¯ X ρ p -quasicontinuous. Thus for each η > U ⊂ ¯ X ρ such that C ¯ X ρ p ( U ) < η and P f | ¯ X ρ \ U is continuous. Setting W = U ∩ Z , we conclude that W is open in Z and that ˇ f | Z \ W is continuous. By Proposition10.7 we have C Z ˇ B θp ( W ) . C ¯ X ρ p ( W ) ≤ C ¯ X ρ p ( U ) < η. Since η > f is C Z ˇ B θp -quasicontinuous. (cid:3) We next consider embeddings of Besov spaces into H¨older spaces. For this next propo-sition we recall the notion of relative lower volume decay defined in (7.17), and recall thatevery doubling measure ν satisfies this condition for Q = log C ν . Proposition 11.4.
Suppose that ν has relative lower volume decay of order Q > . Let p ≥ and < θ < be given. We set β = p (1 − θ ) and let Q β = max { , Q + β } . If p > Q β then every f ∈ ˜ B θp ( Z ) has a ν -a.e. representative that is (1 − Q β /p ) -H¨older continuous oneach ball B ⊂ Z .Proof. Let f ∈ ˜ B θp ( Z ) be given with p satisfying p > Q β . As in the proof of Proposition 11.3we let ˇ f = P f | Z denote the restriction of the extension P f to Z , which satisfies P f | Z = fν -a.e. by Proposition 10.6. By Lemma 7.6 the metric measure space ( ¯ X ρ , d ρ , µ β ) has relativelower volume decay of order Q β . Since p > Q β , we conclude from [21, Theorem 9.2.4] thatfunctions in ˜ D ,p ( ¯ X ρ ) are (1 − Q β /p )-H¨older continuous on each ball ˆ B ⊂ ¯ X ρ . Since themetric d ρ on ¯ X ρ restricts to the metric d on Z , we conclude that P f | Z is (1 − Q β /p )-H¨older continuous on any ball B = B Z ( z, r ) ⊂ Z , viewing this ball as a subset of the ballˆ B = B ρ ( z, r ) ⊂ ¯ X ρ . (cid:3) The calculations after [3, Proposition 13.7] give more precise details about the settingsin which Proposition 11.4 applies.We can also show that functions f ∈ ˜ B θp ( Z ) have Lebesgue points quasieverywhere withrespect to the Besov capacity. For this we will need to assume that the measure ν also XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 65 satisfies a reverse-doubling condition for an exponent η > < r ′ ≤ r ,(11.1) ν ( B Z ( z, r ′ )) ν ( B Z ( z, r )) ≤ C rev (cid:18) r ′ r (cid:19) η , for some constant C rev ≥
1. Under the hypothesis that ν is doubling, the reverse-doublingcondition (11.1) on ν is equivalent to Z being uniformly perfect [25, Lemma 7], which inturn implies that for each ball B ⊂ Z we have(11.2) r ( B ) ≍ diam( B ) , with implied constants depending only on the doubling constant for ν and the constant C rev and exponent η in (11.1).We then have the following two-weighted Poincar´e inequality [3, Proposition 13.5], whichis a special case of an inequality of Bj¨orn-Ka lamajska [5, Theorem 3.1]; we note that thisinequality still applies in our setting since we assume that the reverse-doubling condition(11.1) holds at all scales and since we have that the metric measure spaces ( X ρ , d ρ , µ β )and ( ¯ X ρ , d ρ , µ β ) support a p -Poincar´e inequality. Below we set β = p (1 − θ ) > B = B Z ( z, r ) for z ∈ Z and r >
0. We recall that ˆ B = B ρ ( z, r ) then denotes the ballcentered at z of the same radius in ¯ X ρ . We will allow X to be a hyperbolic filling of Z for any parameter 0 < a < τ > max { , (1 − a ) − } , with the understanding that theimplied constant in (11.3) further depends on a and τ . Proposition 11.5.
Let ≤ p < q < ∞ and u ∈ ˜ N ,p ( ¯ X ρ ) be such that ν -a.e. point of Z is an L ( ¯ X ρ ) -Lebesgue point for u . Let g be a p -integrable p -weak upper gradient for u on ¯ X ρ . Suppose further that ν satisfies the reverse-doubling condition (11.1) . Then for all balls B = B Z ( z, r ) ⊂ Z with z ∈ Z and r > , (11.3) (cid:18)Z B | u − u ˆ B | q dν (cid:19) /q . Θ q ( r ) (cid:18)Z B g p dµ β (cid:19) /p . with implied constant depending additionally on the constant C rev and exponent η > in (11.1) , where Θ q ( r ) = sup
Suppose that ν has relative lower volume decay of order Q > and that ν satisfies the reverse-doubling condition (11.1) for some η > . Let p ≥ and < θ < begiven such that pθ < Q and set Q ∗ = Qp/ ( Q − pθ ) . Let f ∈ ˜ B θp ( Z ) be given. Then there is afunction ˇ f ∈ ˜ B θp ( Z ) such that f = ˇ f ν -a.e. and C Z ˇ B θp -q.e. point z ∈ Z is an L Q ∗ ( Z ) -Lebesguepoint of ˇ f . In particular we have f ∈ L Q ∗ loc ( Z ) . The conclusion means that there is a set G ⊂ Z with C Z ˇ B θp ( Z \ G ) = 0 for which every point z ∈ G is an L Q ∗ ( Z )-Lebesgue point of ˇ f . Note by H¨older’s inequality that the conclusion implies that C Z ˇ B θp -q.e. point z ∈ Z is an L q ( Z )-Lebesgue point of ˇ f for 1 ≤ q ≤ Q ∗ . We alsonote that we always have Q ∗ > p when pθ < Q . Proof.
Throughout this proof all implied constants will be allowed to additionally dependon the constant C rev and exponent η > X of Z for arbitrary 0 < a < τ > max { , (1 − a ) − } , so we will be allowing implied constants to also depend on thoseparameters a and τ .Let f ∈ ˜ B θp ( Z ) be given as in the hypotheses. We let u ∈ ˜ D ,p ( ¯ X ρ ) be any functionsuch that u | Z = f ν -a.e. on Z ; by Proposition 10.6 we can take u = P f , but we will allowfor a more general choice of u in the proof. We then set ˇ f = u | Z . We note that thisfunction u agrees µ β -a.e. with the extension ˆ u of u | X ρ to ¯ X ρ given by Proposition 9.11 since µ β ( ∂X ρ ) = 0, which implies that ˆ u = u C ¯ X ρ p -q.e. since both of these functions belong to˜ D ,p ( ¯ X ρ ). By Proposition 9.11 and H¨older’s inequality we then have for C ¯ X ρ p -q.e. z ∈ Z ,(11.5) lim r → + − Z B ρ ( z,r ) | u − ˇ f ( z ) | dµ β = 0 , where we have used that µ β is extended to Z = ∂X ρ by µ β ( ∂X ρ ) = 0. By Proposition9.9 we then have that ν -a.e. point of Z is an L ( ¯ X ρ )-Lebesgue point for u . Let g be any p -integrable p -weak upper gradient for u on ¯ X ρ . Then by [21, Lemma 9.2.4] we have for C ¯ X ρ p -q.e. z ∈ Z ,(11.6) lim r → + r p − Z B ρ ( z,r ) g p dµ β = 0 . By Proposition 10.7 it follows that each of these assertions also hold for C Z ˇ B θp -q.e. point of Z .We will show that any point z ∈ Z for which (11.5) and (11.6) hold is an L Q ∗ ( Z )-Lebesguepoint of ˇ f .Let z ∈ Z be a point for which (11.5) and (11.6) hold. By Proposition 11.5 applied with q = Q ∗ > p we have for each ball B = B Z ( z, r ) ⊂ Z ,(11.7) (cid:18)Z B | ˇ f − u ˆ B | Q ∗ dν (cid:19) /Q ∗ . Θ Q ∗ ( r ) (cid:18)Z B g p dµ β (cid:19) /p . By combining this with the triangle inequality in L Q ∗ ( Z ) we conclude that (cid:18)Z B | ˇ f − ˇ f ( z ) | Q ∗ dν (cid:19) /Q ∗ ≤ (cid:18)Z B | ˇ f − u ˆ B | Q ∗ dν (cid:19) /Q ∗ + ν ( B ) /Q ∗ | u ˆ B − ˇ f ( z ) | . Θ Q ∗ ( r ) (cid:18)Z B g p dµ β (cid:19) /p + ν ( B ) /Q ∗ Z ˆ B | u − ˇ f ( z ) | dµ β . By dividing through by ν ( B ) /Q ∗ , we conclude that(11.8) (cid:18) − Z B | ˇ f − ˇ f ( z ) | Q ∗ dν (cid:19) /Q ∗ . Θ Q ∗ ( r ) µ β (2 ˆ B ) /p ν ( B ) /Q ∗ (cid:18) − Z B g p dµ β (cid:19) /p + − Z ˆ B | u − ˇ f ( z ) | dµ β . The second term on the right converges to 0 as r → Q ∗ ( r ) µ β (2 ˆ B ) /p ν ( B ) /Q ∗ . r. XTENSION AND TRACE THEOREMS FOR NONCOMPACT DOUBLING SPACES 67
By (11.4), Lemma 7.3, and the lower volume decay bound (7.17) (here we are using anequivalent uncentered version of this bound applied to the containment of balls B Z ( z, s ) ⊂ B for z ∈ B , 0 < s ≤ r , see [21, (9.1.14)]) we haveΘ Q ∗ ( r ) µ β (2 ˆ B ) /p ν ( B ) /Q ∗ . r Q/Q ∗ − Q/p + β/p sup , and the exponent of r in the first line simplifies to QQ ∗ − Qp + βp = Q − pθp − Qp + p (1 − θ ) p = 1 − θ. We conclude that C Z ˇ B θp -q.e. point of Z is an L Q ∗ ( Z )-Lebesgue point of ˇ f , as desired. Lastly,for use in Proposition 11.7 below we note that combining (11.7), (11.9), and the radiusestimate (11.2),(11.10) (cid:18) − Z B | f − u ˆ B | Q ∗ dν (cid:19) /Q ∗ . diam( B ) (cid:18) − Z B g p dµ β (cid:19) /p , where we have used that f = ˇ f ν -a.e. on Z . Here g denotes any p -integrable p -weakupper gradient for the extension u ∈ ˜ D ,p ( ¯ X ρ ), and the implied constant depends on theconstants and exponents in (7.17) and (11.1) as well as the constants a and τ associated tothe hyperbolic filling X and the exponents p and θ . (cid:3) By H¨older’s inequality the inequality (11.10) implies the following important inequalityfor functions in ˜ B θp ( Z ), which we will call a hyperbolic ( q, p ) -Poincar´e inequality since itrelates the q -means of functions on balls in Z to the p -norms of upper gradients of extensionsof those functions to the uniformized hyperbolic filling ¯ X ρ . The second inequality followsfrom the first by applying Lemma 5.1 with α = u ˆ B . Proposition 11.7.
Suppose that ν has relative lower volume decay of order Q > and that ν satisfies the reverse-doubling condition (11.1) for some η > . Let p ≥ and < θ < be given such that pθ < Q and set Q ∗ = Qp/ ( Q − pθ ) . Let f ∈ ˜ B θp ( Z ) be given and let g : ¯ X ρ → [0 , ∞ ] be a Borel function that is a p -integrable p -weak upper gradient of somefunction u ∈ ˜ D ,p ( ¯ X ρ ) such that u | Z = f ν -a.e. Then for any ≤ q ≤ Q ∗ and any ball B ⊂ Z we have (cid:18) − Z B | f − u ˆ B | q dν (cid:19) /q . diam( B ) (cid:18) − Z B g p dµ β (cid:19) /p , with implied constant depending only on the constants and exponents in (7.17) and (11.1) ,the constants a and τ associated to the hyperbolic filling X , and the exponents p and θ .Consequently we have (cid:18) − Z B | f − f B | q dν (cid:19) /q . diam( B ) (cid:18) − Z B g p dµ β (cid:19) /p . References [1] A. Bj¨orn and J. Bj¨orn.
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