Carleson type estimates forp-harmonic functions andthe Conformal Martin boundary of John domains in

metric measure spaces∗

Hiroaki Aikawa†and Nageswari Shanmugalingam‡

Dedicated to Professor Saburou Saitoh on the occasion of his 60th birthday

Abstract

A Carleson type estimate provides control of the bound of positive harmonic functionsvanishing on a portion of the boundary. Such an estimate is well-known for harmonic functionsin certain Euclidean domains. In this note we prove a Carleson type estimate forp-harmonicfunctions on bounded John domains in a complete metric space equipped with an AhlforsQ-regular measure supporting a (1, p)-Poincare inequality for some 1< p ≤ Q. This result isnew even in the Euclidean setting whenp , 2. We then use the Carleson estimate to study theconformal Martin boundary of bounded John domains in metric measure spaces ofQ-boundedgeometry.

1 Introduction

In the study of the local Fatou theorem for harmonic functions, Carleson [Ca] proved the followingcrucial estimate for positive harmonic functions, now referred to as theCarleson estimate. Given abounded Lipschitz domainD in the Euclidean spaceRn there exist constantsK, C > 1, dependingonly onD, with the following property: Ifξ ∈ ∂D, r > 0 sufficiently small, andxr is a point inDwith |xr − ξ| = r and dist(xr , ∂D) ≥ r/C, then

u ≤ Ku(xr) on D ∩ B(ξ, r).

wheneveru is a positive harmonic function inD∩B(ξ,Cr) vanishing continuously on∂D∩B(ξ,Cr).HereB(ξ, r) denotes the open ball with centerξ and radiusr.

The Carleson estimate has been verified for more general Euclidean domains such as NTAdomains, and plays an important role in the study of harmonic analysis on nonsmooth domains.There are at least three different proofs of the Carleson estimate based on: (i) uniform barriers, (ii)the boundary Harnack principle, and (iii) the mean value inequality of subharmonic functions:

∗2000 Mathematics Subject Classification: Primary 31C35; Secondary 31C45, 30C65.Keywords: quasi-minimizers, singular function, conformal map, John domain, conformal Martin boundary.

†H. A. was partially supported by Grant-in-Aid for (B) (No. 15340046) and Exploratory Research (No. 13874023)Japan Society for the Promotion of Science.

‡N. S. was partially supported by NSF grant DMS-0243355.

1

(i) Carleson’s original proof as well as the extension to NTA domains due to Jerison–Kenig [JK]are based on uniform barriers. This method was used also by the second author, together withHolopainen and Tyson, in [HST] to study conformal Martin boundaries of bounded uniformdomains in metric measure spaces of bounded geometry. This approach requires the notionof uniform fatnessof the boundary introduced by Lewis in [L].

(ii) In [Ai1], the first named author proved the boundary Harnack principle directly and verifiedthe Carleson estimate as a corollary. This method does not rely on uniform barriers andis applicable to uniform domains with a small boundary, and more generally even to anirregular uniform domain. However, this method does not seem to be applicable to nonlinearequations.

(iii) In the study of Denjoy domains, Benedicks [Be] employed Domar’s argument [D] basedon the mean value inequality of subharmonic functions. His approach was generalized toLipschitz Denjoy domains by Chevallier [Chv]. The first author, together with Hirata andLundh, utilized Domar’s argument in [AHL] to prove a version of the Carleson estimate forJohn domains in Euclidean spaces.

The first goal of this note is to show that Domar’s argument applies not only to harmonicfunctions on an Euclidean domain but also to solutions of certain nonlinear equations on metricmeasure spaces. Throughout this paper, we assume that (X,d, µ) is a proper metric measure spacewith at least two points and thatµ a doubling Borel measure. Here we say thatX is proper if closedand bounded subsets ofX are compact; and thatµ is doubling if there is a constantCd ≥ 1 suchthat

µ(B(x,2r)) ≤ Cdµ(B(x, r)),

whereB(x, r) = y ∈ X : d(x, y) < r is the open ball with centerx and radiusr. Moreover, wefix 1 < p < ∞, and assume thatX supports a (1, p)-Poincare inequality (see Definition2.1below).We shall establish a Carleson type estimate (Theorem5.2 below) for John domains in the settingof such metric measure spaces by adapting the version of Domar’s argument found in [AHL].

Our second goal is the study of conformal Martin boundaries of bounded John domains whoseboundaries may not be uniformly fat. Under the additional assumption that the measureµ isAhlfors Q-regular (for someQ > 1), we will use the above Carleson type estimate to extendthe results of [HST] and [Sh3] to greater generality. One of our main results is Theorem6.1, whichdescribes the behavior of the conformal Martin kernels. The growth estimate (Theorem6.1(ii )) isnew.

In the general setting of metric measure spaces, it is not clear whether there exists even onebounded uniform domain inX. However, ifX is a geodesic space, then every ball inX is a Johndomain with the center of the ball acting as a John center. It is a well-known fact that any doublingmetric measure space supporting a (1, p)-Poincare inequality is quasi-convex; that is, there is aconstantq ≥ 1 such that for every pair of pointsx, y ∈ X there is a rectifiable curveγxy in Xconnectingx to y with the property that the lengthℓ(γxy) of γxy satisfies

ℓ(γxy) ≤ q d(x, y). (1)

Thus in our situation there are a plethora of bounded John domains inX even ifX is not a geodesicspace. It is therefore desirable to study the conformal Martin boundary of bounded John domains

2

in X. The theory developed in [HST] indicates that the conformal Martin boundary is conformallyinvariant. The results developed in this note are therefore useful in the study of a Fatou typeproperty of conformal mappings between two bounded John domains in metric spaces; see [Sh3].

Acknowledgement: The authors wish to thank the referee for helpful suggestions on improvingthe exposition of this paper.

2 Definitions and Notations

Unless otherwise stated,C denotes a positive constant whose exact value is unimportant, canchange even within the same line, and depends only on fixed parameters such asX, d, µ andp. If necessary, we will specify its dependence on other parameters.

In the setting of metric measure spaces that may not have a Riemannian structure, the followingnotion of upper gradients, first formulated by Heinonen and Koskela in [HeK1], replaces the notionof distributional derivatives (in [HeK1] upper gradients are referred to as very weak gradients). ABorel functiong on X is anupper gradientof a real-valued functionf on X if for all non-constantrectifiable pathsγ : [0, lγ] → X parameterized by arc length,

| f (γ(0))− f (γ(lγ))| ≤∫γ

g ds,

where the above inequality is interpreted as saying also that∫γ

g ds= ∞ whenever at least one of| f (γ(0))| and| f (γ(lγ))| is infinite. See [HeK1] and [KoMc] for more on this notion.

Definition 2.1. We say thatX supports a (1, p)-Poincare inequalityif there are constantsκ ≥ 1 andCp ≥ 1 such that for all ballsB(x, r) ⊂ X, all measurable functionsf on X, and allp-weak uppergradientsg of f , ∫

B(x,r)

| f − fB(x,r)|dµ ≤ Cpr( ∫

B(x,κr)

gp dµ)1/p

,

where

fB(x,r) :=∫B(x,r)

f dµ :=1

µ(B(x, r))

∫B(x,r)

dµ.

Following [Sh1], we consider a version of Sobolev spaces onX.

Definition 2.2. Let

∥u∥N1,p =

(∫X|u|p dµ

)1/p

+ infg

(∫X

gp dµ)1/p

,

where the infimum is taken over all upper gradients ofu. TheNewtonian spaceonX is the quotientspace

N1,p(X) = u : ∥u∥N1,p < ∞/∼,whereu ∼ v if and only if ∥u− v∥N1,p = 0.

3

The spaceN1,p(X) equipped with the norm∥ · ∥N1,p is a Banach space and a lattice, see [Sh1].An alternative definition of Sobolev spaces given by Cheeger in [Ch] yields the same space asN1,p(X) wheneverp > 1; see Theorem 4.10 in [Sh1]. Cheeger’s definition yields the notion ofpartial derivatives in the following theorem (Theorem 4.38 in [Ch]).

Theorem 2.3 (Cheeger). Let X be a metric measure space equipped with a positive doublingBorel regular measureµ admitting a(1, p)-Poincare inequality for some1 < p < ∞. Then thereexists a countable collection(Uα,Xα) of measurable sets Uα and Lipschitz “coordinate” functionsXα : X→ Rk(α) such thatµ

(X \∪αUα

)= 0, and for eachα the following conditions hold.

The measure of Uα is positive and1 ≤ k(α) ≤ N, where N is a constant depending only on thedoubling constant ofµ and the constant from the Poincare inequality. If f: X → R is Lipschitz,then there exist unique bounded measurable vector-valued functions dα f : Uα → Rk(α) such thatfor µ-a.e. x0 ∈ Uα,

limr→0+

supx∈B(x0,r)

| f (x) − f (x0) − dα f (x0) · (Xα(x) − Xα(x0))|r

= 0.

We can assume that the setsUα are pairwise disjoint, and extenddα f by zero outsideUα.Regardingdα f (x) as vectors inRN, we letd f =

∑α dα f . The differential mappingd : f 7→ d f is

linear and it is shown in page 460 of [Ch] that there is a constantC > 0 such that for all Lipschitzfunctions f andµ-a.e.x ∈ X,

1C|d f(x)| ≤ gf (x) := inf

glim sup

r→0+

∫B(x,r)

g dµ ≤ C|d f(x)|. (2)

Here|d f(x)| is a norm coming from a measurable inner product on the tangent bundle ofX createdby the above Cheeger derivative structure (see the discussion in [Ch]), and the infimum is takenover all upper gradientsg ∈ Lp(X) of f ; gf is in some sense theminimalupper gradient off (seeCorollary 3.7 in [Sh2]). Also, by Proposition 2.2 in [Ch], d f = 0 µ-a.e. on every set wheref isconstant.

By [Ch, Theorem 4.47] or [Sh1, Theorem 4.1], the Newtonian spaceN1,p(X) is equal to theclosure in theN1,p-norm of the collection of Lipschitz functions onX with finite N1,p-norm. By[FHK, Theorem 10], there exists a unique “gradient”du satisfying (2) for every u ∈ N1,p(X).Moreover, if uj∞j=1 is a sequence inN1,p(X), thenuj → u in N1,p(X) if and only if as j → ∞,uj → u in Lp(X, µ) andduj → du in Lp(X, µ;RN). Hence the differential structure extends to allfunctions inN1,p(X). Throughout this note we will use this structure, see for example Definition2.5below.

Definition 2.4. The p-capacityof a Borel setE ⊂ X is the number

Capp(E) := infu

(∫X|u|p dµ +

∫X|du|p dµ

),

where the infimum is taken over allu ∈ N1,p(X) such thatu = 1 on E. A property is said to holdp-quasieverywhere inX if the set on which the property does not hold has zerop-capacity. Therelative p-capacityCapp(K;Ω) of a compact setK with respect to an open setΩ ⊃ K is given by

Capp(K;Ω) = inf∫Ω

|du|pdµ,

4

where the infimum is taken over all functionsu ∈ N1,p(X) for which u|K ≥ 1 andu|X\Ω = 0. Ifno such function exists, then we set Capp(K;Ω) = ∞. For more on capacity, see [AO], [HeK2],[KiMa0], [HKM, Chapter 2], and the references therein.

To compare the boundary values of Newtonian functions, we need a Newtonian space with zeroboundary values. LetΩ ⊂ X be an open set and let

N1,p0 (Ω) = u ∈ N1,p(X) : u = 0 p-quasieverywhere onX \Ω.

Corollary 3.9 in [Sh1] implies thatN1,p0 (Ω) equipped with theN1,p-norm is a closed subspace of

N1,p(X). By [Sh2, Theorem 4.8], ifΩ is relatively compact, then the space Lipc(Ω) of Lipschitzfunctions with compact support inΩ is dense inN1,p

0 (Ω). In the rest of this paper,Ω ⊂ X willalways denote a bounded domain inX with Capp(X \Ω) > 0.

Definition 2.5. LetΩ ⊂ X be a domain. A functionu : X→ [−∞,∞] is said to bep-harmonicinΩ if u ∈ N1,p

loc (Ω) and for all relatively compact subsetsU of Ω and for every functionφ ∈ N1,p0 (U),∫

U|du|p dµ ≤

∫U|d(u+ φ)|p dµ. (3)

We say thatu is a p-subsolution inΩ if (3) holds for every non-positive functionφ ∈ N1,p0 (U).

We say thatu is ap-quasiminimizerif there is constantCqm ≥ 1 such that for all relatively compactsubsetsU of Ω and every functionφ ∈ N1,p

0 (U),∫U|du|p dµ ≤ Cqm

∫U|d(u+ φ)|p dµ. (4)

Furthermore, we say thatu is a p-quasisubminimizerif (4) holds true wheneverφ is a non-positivefunction inN1,p

0 (U).

Remark 2.6. It is easily seen thatp-harmonic functions arep-quasiminimizers and thatp-subsolutionsarep-quasisubminimizers. See [KiMa2] and [KiSh] for more on quasiminimizers.

Definition 2.7. By HUp f we denote the solution to thep-Dirichlet problem on the open setU with

boundary dataf ∈ N1,p(U), i.e., HUp f is p-harmonic inU and HU

p f − f ∈ N1,p0 (U). An upper

semicontinuous functionu is said to bep-subharmonic inΩ if the comparison principle holds, i.e.,if f ∈ N1,p(U) is continuous up to∂U andu ≤ f on ∂U, thenu ≤ HU

p f on U for all relativelycompact subsetsU of Ω.

Definition 2.8. LetΩ be a relatively compact domain inX and lety ∈ Ω. An extended real-valuedfunctiong = g(·, y) onΩ is said to be ap-singular function with singularity at yif it satisfies thefollowing four criteria:

(i) g is p-harmonic inΩ \ y andg > 0 onΩ;

(ii) g∣∣∣X\Ω = 0 p-q.e. andg ∈ N1,p(X \ B(y, r)) for eachr > 0;

(iii) y is a singularity, i.e., limx→y g(x) = ∞;

5

(iv) whenever 0≤ a < b < ∞,Capp(Ω

b;Ωa) = (b− a)1−p, (5)

whereΩb = x ∈ Ω : g(x) ≥ b andΩa = x ∈ Ω : g(x) > a.

In [HoSh] it was shown that every relatively compact domain in a metric measure spaceequipped with a doubling measure supporting a (1,q)-Poincare inequality withq < p has ap-singular function which plays a role analogous to the Green function of the Euclideanp-Laplaceoperator. It was shown by Keith–Zhong in [KZ] that a complete metric space supporting a (1, p)-Poincare inequality for somep > 1 also supports a (1,q)-Poincare inequality for some 1≤ q < p.Hence to apply the results of [KiSh] and [HoSh] we do not need to apriori assume the betterPoincare inequality.

Note that we have the doubling property on the measureµ as a standing assumption. As aconsequence of this doubling property, it can be shown that there are constantsQ > 0 andC1 suchthat for allx ∈ X, 0 < ρ < R, andy ∈ B(x,R),

1C1

(ρ

R

)Q

≤ µ(B(y, ρ))µ(B(x,R))

. (6)

The book [He] has a proof of this fact. The measureµ is said to beAhlfors Q-regularif there is aconstantC ≥ 1 such that for everyx ∈ X and for everyr > 0,

rQ

C≤ µ(B(x, r)) ≤ C rQ. (7)

For the rest of this section we will assume thatX = (X,d, µ) is of Q-bounded geometry, i.e.,µ is Ahlfors Q-regular andX supports a (1,Q)-Poincare inequality ([BHK, Section 9] or [HST]).It was shown in [HeK1] that metric spaces ofQ-bounded geometry possess a Loewner type prop-erty related to theQ-modulus of curve families connecting compacta. Therefore we can use thetechniques of [Ho] to show that for eachy ∈ Ω there is exactly oneQ-singular function forΩ withsingularity aty satisfying equation (5). This enables us to define a boundary in a manner similar tothe classical potential theoretic Martin boundary.

Definition 2.9. Fix x0 ∈ Ω. Given a sequence (xn) of points inΩ, we say that the sequence isfundamental(relative tox0) if it has no accumulation point inΩ and the sequence of normalizedsingular functions

Mxn(x) = M(x, xn) :=g(x, xn)g(x0, xn)

is locally uniformly convergent inΩ. Hereg is theQ-singular function forΩ. We setM(x, x0) = 0whenx , x0, and setM(x0, x0) = 1.

Given a fundamental sequenceξ = (xn), we denote the corresponding limit function

M(x) = Mξ(x) := limn→∞

M(x, xn),

and call it aconformal Martin kernelfunction. We say that two fundamental sequencesξ andζare equivalent (relative tox0), ξ ∼ ζ, if Mξ = Mζ. It is worth noting thatMξ is a non-negative

6

Q-harmonic function inΩ, with Mξ(x0) = 1. HenceMξ > 0 in Ω by local Harnack’s inequal-ity (see [KiSh, Corollary 7.3]). Note that if ˜x0 is another point inΩ, theng(x, xn)/g(x0, xn) =M(x, xn)/M(x0, xn). Therefore, the property of being a fundamental sequence is independent of theparticular choice ofx0. Furthermore, fundamental sequencesξ andζ are equivalent relative tox0 ifand only if they are equivalent relative to any ˜x0 ∈ Ω. Thus the following definition is independentof the fixed pointx0.

Given a pointχ ∈ ∂Ω we say that a functionM is a conformal martin kernel associated withχif there is a fundamental sequence (yn) in Ω so thatyn → χ and the sequence of singular functionsM(·, yn) with singularity atyn converges locally uniformly toM.

Definition 2.10. The collection of all equivalence classes of fundamental sequences inΩ (or equiv-alently, the collection of all conformal Martin kernel functions) is theconformal Martin boundary∂cMΩ of the domainΩ. This collection is endowed with the local uniform topology: a sequenceξnin this boundary is said to converge to a pointξ if the sequence of functionsMξn converges locallyuniformly to Mξ.

The classical Martin boundary theory can be extended to general domains in metric measurespaces under certain circumstances. However, there are examples of metric measure spaces withAhlfors Q-regular measure,Q > 2, supporting a (1,Q)-Poincare inequality but not (1,2)-Poincareinequality. For domains in such a metric space, 2-singular functions of Definition2.8 may notexist and it is not easy to say what kind of ideal boundary should correspond to the classicalMartin boundary, whereas the conformal Martin boundary can be constructed immediately as in[HST].

3 Domar’s argument

Recall thatX is a proper metric space and thatµ is doubling and supports a (1, p)-Poincare inequal-ity. In this section we assume thatΩ ⊂ X is a bounded open set. Increasing the value ofC1 in (6)to absorb the terms involvingB(x,2 diam(Ω)) and (2 diam(Ω))Q, we obtain the lower mass bound:

µ(B(y, r)) ≥ rQ

C1for y ∈ Ω and 0< r ≤ 2 diam(Ω). (8)

Let u be a non-negative, locally boundedp-subharmonic function or ap-quasisubminimizer inΩ. Thenu is a p-quasisubminimizer (see [KiMa1, Corollary 7.8]) and henceu is in the De GiorgiclassDGp(Ω) (see [KiMa2, Lemma 5.1]). This means that ifB(x,R) ⊂ Ω, then∫

y∈B(x,ρ) : u(y)>kgp

u dµ ≤ C(r − ρ)p

∫y∈B(x,r) : u(y)>k

(u− k)p dµ

for everyk ∈ R and 0< ρ < r < R/κ, whereκ is the scaling constant from the Poincare inequality.It therefore follows from [KiSh, Theorem 4.2] (withk0 = 0) and the doubling property of themeasureµ that if B(x,R) ⊂ Ω, then

u(x) ≤ Cs

( ∫B(x,R)

up dµ

)1/p

, (9)

7

whereCs ≥ 1 is independent ofx, R and u, but depends on the quasisubminimizing constantCqm. If a functionu on an open setU satisfies (9) for every ballB(x,R) ⊂ U, then we say thatu enjoys theweak sub-mean value property in U. Using the weak sub-mean value property, weshall give the following modification of Domar’s theorem (see [D] and [AHL]). Observe that theweak sub-mean value property (9) holds for more general classes of functions than the class ofp-harmonic functions. Indeed, [KiSh] proved this property forp-quasiminimizers and more generallyfor functions in the De Giorgi class; see [KiSh] and [KiMa2]. Hence the following lemma isphrased for the class ofall functions satisfying the weak sub-mean value property, though, in thisnote, it will later be applied only top-quasisubminimizers. Foru > 0 we write

log+ u =

logu if u ≥ 1,

0 otherwise.

Lemma 3.1. Let Ω be a bounded open set in X and letδΩ(x) = dist(x,X \ Ω). Suppose u is alocally bounded non-negative function onΩ satisfying the weak sub-mean value property (9) in Ω.If there is a positive real numberε with

I :=∫Ω

(log+ u)Q−1+ε dµ < ∞,

where Q is the lower mass bound in(8), then there exists a constant C> 0 independent of u suchthat

u(x) ≤ 4C2s exp(CI1/εδΩ(x)−Q/ε) for all x ∈ Ω. (10)

Note that in the Euclidean setting we haveδΩ(x) = dist(x, ∂Ω), but in the general setting ofmetric measure spaces this may not be the case.

Proof. We first prove the following estimate.Let x ∈ X andR> 0 such thatu satisfies thep-submeanvalue property onB(x,R). If u(x) ≥ t >

0, a ≥ 2Cs, and

µ(y ∈ B(x,R) :ta< u(y) ≤ at) ≤ µ(B(x,R))

a2p, (11)

then there existsx2 ∈ B(x,R) such thatu(x2) > at. HereCs is the constant in (9). To see this,suppose thatu(y) ≤ at for everyy ∈ B(x,R). Then by (9),

t ≤ u(x) ≤ Cs

∫

B(x,R)

up dµ

1/p

≤ Cs

(1

µ(B(x,R))

[( ta

)p

µ(B(x,R)) +∫y∈B(x,R) : u(y)>t/a

up dµ

])1/p

≤ 21/pCs

at.

SinceCs/a ≤ 1/2 andp > 1 by assumption, we have 1≤ 21/pCs/a < 1, which is not possible.Hence there must be a pointx2 ∈ B(x,R) such thatu(x2) > at.

Whatever the value ofC might be, we haveC > 0. Hence exp(CI1/εδΩ(x)−Q/ε) ≥ 1. Thereforein order to prove (10), it suffices to show that there exists a constantC > 0 such that wheneveru(x) > 4C2

s,u(x) ≤ exp(CI1/εδΩ(x)−Q/ε). (12)

8

Fix x ∈ Ω such thatu(x) > 4C2s. Thenu(x) > 1 and hence log+ u(x) = logu(x). Therefore,

demonstrating (12) is equivalent to showing that there is a constantC > 0 independent ofx so that

δΩ(x) ≤ CI1/Q (log+ u(x)

)−ε/Q. (13)

To this end, let us choosea = 2Cs. This sets us up to use (11). For j ∈ N let

Rj =[C1 a2pµ(y ∈ Ω : aj−2u(x) < u(y) ≤ aju(x))

]1/Q,

whereC1 is the constant in (8). We claim

δΩ(x) ≤ 2∞∑j=1

Rj . (14)

In order to prove (14), we now construct a sequence of points inΩ, finite or infinite dependingon the situation, as follows. Letx1 = x. If δΩ(x1) < R1, then consider the singleton sequence (x1).SupposeδΩ(x1) ≥ R1. SinceB(x1,R1) ⊂ Ω, it follows that[

C1 a2pµ(y ∈ B(x1,R1) : a−1u(x1) < u(y) ≤ au(x1))]1/Q≤ R1.

It therefore follows from (8) that

µ(y ∈ B(x1,R1) : a−1u(x1) < u(y) ≤ au(x1)) ≤RQ

1

C1 a2p≤ µ(B(x1,R1))

a2p.

Now, by (11) with t = u(x1), there is a pointx2 ∈ B(x1,R1) such thatu(x2) > au(x1). If δΩ(x2) < R2,then consider the sequence (x1, x2). Otherwise,B(x2,R2) ⊂ Ω, and as before we can apply (11)with t = au(x1) to get x3 ∈ B(x2,R2) such thatu(x3) > au(x2) > a2u(x1). Inductively, we mayconstructxJ given (x1, x2, . . . , xJ−1) such that forj = 1, . . . , J − 1,

δΩ(xj) ≥ Rj , d(xj , x j−1) < Rj−1, and u(x j) > au(x j−1) > aj−1u(x1).

If δΩ(xJ−1) < RJ−1, then we stop here. Otherwise, we may use (11) to find a pointxJ ∈ B(xJ−1,RJ−1)such thatu(xJ) > au(xJ−1) > aJ−1u(x1). We will now show thatδΩ(x1) ≤ 2

∑∞j=1 Rj. To do so, we

consider two cases.Case 1:the sequence is finite. Then there is a positive integerJ such thatδΩ(xJ) < RJ. Then

δΩ(x1) ≤J−1∑j=1

d(x j , xj+1) + δΩ(xJ) <J−1∑j=1

Rj + RJ ≤∞∑j=1

Rj .

Case 2: the sequence is infinite. Then for everyj ∈ N we haveu(x j) > aj−1u(x1); that is,lim j→∞ u(xj) = ∞. But then asu is locally bounded onΩ, the infinite sequence (x j) j has noaccumulation point inΩ. Since, by assumption,X is proper and henceΩ is compact, there is asubsequence converging to a point in∂Ω. Hence there exists someJ ∈ N for which δΩ(xJ) <12δΩ(x1). Hence

δΩ(x1) ≤J−1∑j=1

d(x j , xj+1) + δΩ(xJ) <J∑

j=1

Rj +12δΩ(x1).

9

we can then conclude that

δΩ(x1) ≤ 2J∑

j=1

Rj ≤ 2∞∑j=1

Rj .

Hence (14) follows whether the sequence constructed above is finite or not.Therefore to show (13) it suffices to prove that

∞∑j=1

Rj ≤ CI1/Q (log+ u(x)

)−ε/Q. (15)

Let j0 be the unique positive integer such thataj0 < u(x) ≤ aj0+1. Then j0 ≥ 2, since we haveassumed thatu(x) > 4C2

s = a2. Recall that forj ∈ N,

Rj =[C1 a2pµ(y ∈ Ω : aj−2u(x) < u(y) ≤ aju(x))

]1/Q.

We obtain from Holder’s inequality that

∞∑j=1

Rj ≤ C∞∑j=1

[µ(y ∈ Ω : aj0+ j−2 < u(y) ≤ aj0+ j+1)

]1/Q

≤ C∞∑

j= j0−1

[µ(y ∈ Ω : aj < u(y) ≤ aj+3)

]1/Q

= C∞∑

j= j0−1

j(Q−1+ε)/Q

j(Q−1+ε)/Q

[µ(y ∈ Ω : aj < u(y) ≤ aj+3)

]1/Q

≤ C

∞∑j= j0−1

1j(Q−1+ε)/(Q−1)

(Q−1)/Q ∞∑

j= j0−1

∫y∈Ω : a j<u(y)≤a j+3

jQ−1+ε dµ

1/Q

≤ C j−ε/Q0

∞∑j= j0−1

∫y∈Ω : a j<u(y)≤a j+3

(log+ uloga

)Q−1+ε

dµ

1/Q

.

Note that eachy ∈ Ω belongs to at most three of the setsy ∈ Ω : aj < u(y) ≤ aj+3. Hence we seethat

∞∑j=1

Rj ≤ C j−ε/Q0 I1/Q.

By the choice ofj0 it can be seen thataj0 < u(x) ≤ aj0+1 and j0 ≥ 2. We therefore havej0 ≤logu(x)/ loga ≤ j0 + 1 ≤ 2 j0, and hence

∞∑j=1

Rj ≤ C

(log+ u(x)2 loga

)−ε/QI1/Q,

which is (15). Note thatC is independent ofx andu. This completes the proof of Lemma3.1.

It should be emphasized that we do not require in the above two lemmata thatΩ be a domain.It is sufficient to assume merely thatΩ be a bounded open subset ofX such thatX \Ω , ∅.

10

4 Geometry of bounded John domains inX

Let Ω ⊂ X be a domain. For 0< c < 1 a rectifiable curveγ connectingx, y ∈ Ω is said to bea c-John curvein Ω if δΩ(z) ≥ cℓ(γxz) for everyz ∈ γ, whereγxz is the subcurve ogγ having xandz as its two endpoints. We say thatΩ is a John domainwith John center x0 ∈ Ω andJohnconstant cif every point x ∈ Ω can be connected tox0 by a c-John curve inΩ. For A > 1, arectifiable curveγ connectingx, y ∈ Ω is said to be aA-uniform curvein Ω if ℓ(γ) ≤ Ad(x, y) andminℓ(γxz, ℓ(γzy) ≤ AδΩ(z) for everyz ∈ γ. We say thatΩ is anA-uniform domainif every pair ofdistinct pointsx, y ∈ Ω can be joined by anA-uniform curveγ in Ω. Obviously, a uniform domainis a John domain; but the converse is not necessarily true.

Givenx ∈ X andR> 0 we letB(x,R) = y ∈ X : d(x, y) ≤ R. Note that in general this may bea larger set than the closure of the open ballB(x,R) itself. By S(x,R) we mean the sphere centeredat x of radiusR: S(x,R) = y ∈ X : d(x, y) = R. This set contains∂B(x,R), but in general couldbe a larger set.

For domainsV ⊂ X, we letkV denote the following quasi-hyperbolic “metric” onV. If x, y ∈ V,then

kV(x, y) = infγ

∫γ

ds(t)δV(γ(t))

,

where the infimum is taken over all rectifiable curves connectingx to y in V. If no such curve existsin V, then we setkV(x, y) = ∞. It is worth noting that ifV is a John domain, thenkV is alwaysfinite-valued; that is, it is indeed a metric. In fact, ifγ is a c-John curve connectingx to y in Ω,then

kΩ(x, y) ≤∫γ

ds(t)δΩ(γ(t))

≤∫ δΩ(x)/2

0

dsδΩ(x)/2

+

∫ ℓ(γ)

δΩ(x)/2

dscs≤ 1+

log 2/cc+

1c

log(δΩ(y)δΩ(x)

). (16)

Suppose for a moment thatΩ is a uniform domain. Letξ ∈ ∂Ω and x ∈ Ω ∩ B(ξ,R/2). By theuniformity we find a pointy ∈ S(ξ,R) with δΩ(y) ≥ R/(3A) and a uniform curveγ ⊂ Ω connectingx andy. We observe thatγ ⊂ Ω ∩ B(ξ,CR) with C > 1 depending only on the uniformity. For thispointy we have

kΩ(x, y) = kΩ∩B(ξ,2CR)(x, y) ≤ C

[log

(RδΩ(x)

)+ 1

].

This property can be generalized to a John domain. Following [AHL], we give the followingdefinition.

Definition 4.1. Let N ∈ N. We say thatξ ∈ ∂Ω has asystem of local reference points of order Nif there exist constantsRξ > 0, λξ > 1 andAξ > 1 such that whenever 0< R < Rξ we can findy1, . . . , yN ∈ Ω ∩ S(ξ,R) with the properties that

(i) A−1ξ R≤ δΩ(yi) ≤ R for eachi = 1, . . . ,N,

(ii) for everyx ∈ Ω ∩ B(ξ,R/2) we have somei ∈ 1, . . . ,N for which

kΩ(x, yi) = kΩ∩B(ξ,λξR)(x, yi) ≤ Aξ

[log

(RδΩ(x)

)+ 1

].

11

Remark 4.2. If Ω is a uniform domain, then everyξ ∈ ∂Ω has a system of local reference pointsof order 1. The constantsRξ, λξ andAξ depend only on the uniformity ofΩ.

Lemma 4.3. LetΩ ⊂ X be a John domain with John center x0 and John constant c. Then thereexists N0 ∈ N depending only on x0, c and the data of X such that everyξ ∈ ∂Ω has a system oflocal reference points of order at most N0 with constants Rξ = R0 := minδΩ(x0)/2,diam(Ω)/10,λξ = 8 and Aξ = A0 := max2/c, 3/2+ c−1 log 2/c.

Proof. Let ξ ∈ ∂Ω and 0< R < R0 and supposex ∈ Ω ∩ B(ξ,R/2). Let γx be ac-John curveconnectingx to x0 and letyx be the point ofγx at whichγx leaves the ballB(ξ,R) for the first time.Then

R≥ δΩ(yx) ≥ cℓ(βx) ≥ cR/2 ≥ R/A0, (17)

whereβx is the subcurve ofγx that terminates atyx. Let E = yx : x ∈ Ω ∩ B(ξ,R/2). By the5-covering lemma (see [He] for example) and the doubling property ofµ we find finitely manypointsy1 = yx1, . . . , yN = yxN ∈ E such thatN ≤ N0,

E ⊂N∪

i=1

B(yi , cR/(10q)) ⊂ B(ξ,R)

andB(yi , cR/(50q))Ni=1 is pairwise disjoint, whereq ≥ 1 is the quasi-convexity constant in (1).Let us demonstrate thaty1, . . . , yN satisfy the conditions in Definition4.1. Obviously, (i) holds

true by (17). For the proof of (ii) takex ∈ Ω ∩ B(ξ,R/2), and the pointyx and the subcurveβx asabove. Then,δΩ∩B(ξ,8R)(z) = δΩ(z) for z ∈ βx, so that (16) gives

kΩ∩B(ξ,8R)(x, yx) ≤∫βx

ds(t)δΩ(βx(t))

≤ 1+log 2/c

c+

1c

log( RδΩ(x)

).

Sinceyx ∈ E ⊂ ∪iB(yi , cR/(10q)), we can find a positive integeri ≤ N such thatyx ∈ B(yi , cR/(10q)).By the quasi-convexity ofX there is a rectifiable curveγ connectingyx to yi with ℓ(γ) ≤ q d(yx, yi) ≤cR/10. By (17),

δΩ∩B(ξ,8R)(z) = δΩ(z) ≥ δΩ(yx) − cR/10> cR/5 for z ∈ γ,

so that

kΩ∩B(ξ,8R)(yx, yi) ≤∫γ

ds(t)δΩ(γ(t))

≤ cR/10cR/5

=12.

Hence

kΩ∩B(ξ,8R)(x, yi) ≤ kΩ∩B(ξ,8R)(x, yx) + kΩ∩B(ξ,8R)(yx, yi) ≤ A0

[log

(RδΩ(x)

)+ 1

].

This completes the proof of Lemma4.3.

Lemma 4.4. LetΩ ⊂ X be a bounded John domain with John center x0 and John constant c. Thenthere exist positive constants C andτ depending only on the data X and ofΩ such that for eachξ ∈ ∂Ω and0 < R< 2cδΩ(x0)/(10q),∫

Ω∩B(ξ,R)

(RδΩ(x)

)τdµ(x) ≤ Cµ(B(ξ,R)).

12

Hajłasz and Koskela [HaK1, Lemma 6] demonstrated the lemma for Euclidean domains by anindirect proof using Sobolev extension and embedding theorems (see [HaK1, Remark 11]). Thefollowing proof (a modification of the proof from [AHL]) is more geometric, and holds true for allrelatively compact John domains in quasiconvex metric measure spaces equipped with a doublingmeasure, irrespective of whetherX supports a Poincare inequality.

Proof. For each nonnegative integerj let E j =∪∞

i= j+1 Vi with

Vi = x ∈ Ω ∩ B(ξ,R+ (q+ 1)22−iR/c) : 2−(i+1)R≤ δΩ(x) < 2−iR.We claim that for everyx ∈ E j there exist two pointsyx, y′x ∈ Ω such that

(i) x ∈ B(yx, (q+ 1)2− jR/c),

(ii) B(y′x,2−( j+2)R) ⊂ Vj ∩ B(yx, (q+ 1)2− jR/c).

To see this letx ∈ E j. Let γx be ac-John curve connectingx and the John centerx0. Observe thatwe can chooseyx ∈ γx such thatδΩ(yx) = 2− jR≥ c d(x, yx), thus satisfying (i). AsX is proper, thereis a pointy∗x ∈ X \Ω with δΩ(yx) = d(yx, y∗x). SinceX is quasi-convex, there is a curveβ connectingyx andy∗x such thatℓ(β) ≤ q2− jR. Let y′x ∈ β ∩ Ω be a point satisfyingδΩ(y′x) = (2−( j+1) + 2− j)R/2.Then forz ∈ B(y′x,2

−( j+2)R),

d(yx, z) ≤ d(yx, y′x) + d(y′x, z) < ℓ(β) + 2−( j+2)R<

(q+ 1)2− jRc

,

d(ξ, z) ≤ d(ξ, x) + d(x, yx) + d(yx, z)

< R+(q+ 1)21− jR

c+

2− jRc+

(q+ 1)2− jRc

= R+(3q+ 4)22− jR

4c,

2−( j+1)R< δΩ(y′x) − d(y′x, z) ≤ δΩ(z) ≤ δΩ(y′x) + d(y′x, z) < 2− jR.

The above three inequalities together yield (ii).In view of (i), the collectionB(yx, (q + 1)2− jR/c)x∈E j forms a covering ofE j. By the 5-

covering lemma we have a pairwise disjoint sub-collectionB(yk, (q+1)2− jR/c)k∈N such thatE j ⊂B(yk,5(q + 1)2− jR/c)k∈N. Sinceµ is a doubling measure, we can findC2 > 1, which dependssolely onCd, such that

µ(B(yk,5(q+ 1)2− jR/c)) ≤ C2 µ(B(y′k,2−( j+2)R)) ≤ C2 µ(Vj ∩ B(yk, (q+ 1)2− jR/c)).

Herey′k is associated withyk in the same manner thaty′x is paired withyx, and the second inequalityabove follows from (ii). Therefore,

∞∑i= j+1

µ(Vi) = µ(E j

)≤

∑k∈Nµ(B(yk,5(q+ 1)2− jR/c)

)≤ C2

∑k∈Nµ(Vj ∩ B(yk, (q+ 1)2− jR/c)

)≤ C2 µ(Vj).

Let 1 < t < 1 + 1/C2 andτ = log2 t. Then proceeding as in [AHL, Proof of Lemma 6] or [Ai1,Lemma 5] we can deduce from the above inequality that∫

Ω∩B(ξ,R)

(RδΩ(z)

)τdµ(z) ≤

∞∑j=0

∫V j

(RδΩ(z)

)τdµ(z) ≤

∞∑j=0

t j+1µ(Vj) ≤ Cµ(B(ξ,R)).

13

Definition 4.5. A finite collectionB(xi , r i)ki=1 of balls is said to be aHarnack chainconnectingtwo pointsx, y ∈ Ω with length kif

(i) for i = 1, . . . , k− 1, B(xi , r i) ∩ B(xi+1, r i+1) , ∅,

(ii) for i = 1, . . . , k, B(xi ,2κr i) ⊂ Ω,

(iii) x ∈ B(x1, r1) andy ∈ B(xk, rk),

whereκ is the scaling constant from the Poincare inequality (see Definition2.1).

In light of the definition ofkΩ(x, y) we observe thatx andy are connected by a Harnack chainof length no more than 1+ CkΩ(x, y). Hence the Harnack inequality forp-quasiminimizers (see[KiSh]) gives the following.

Lemma 4.6. If h is a positive p-quasiminimizer onΩ, then

1AH

exp(−AH kΩ(x, y)) ≤ h(x)h(y)

≤ AH exp(AH kΩ(x, y)) for x, y ∈ Ω,

where AH > 1 is independent of h, x and y.

5 A Carleson type estimate forp-harmonic functions on a Johndomain in Ahlfors regular spaces

We will henceforth assume thatµ is Ahlfors Q-regular (see (7)). We will also assume from now onthatΩ ⊂ X is a bounded John domain with John centerx0 ∈ Ω and John constant 0< c < 1 andthatX \Ω , ∅. If x ∈ Ω andγx is a John curve connectingx to a local reference pointyi associatedwith x as in the proof of Lemma4.3, then

kΩ(x, yi) ≤ C3

[log

(δΩ(yi)δΩ(x)

)+ 1

].

Therefore by Lemma4.6, for every positivep-quasiminimizerh onΩ andx ∈ Ω we have

h(x)h(yi)

≤ AH exp

(AHC3

[log

(δΩ(yi)δΩ(x)

)+ 1

])≤ C

(δΩ(yi)δΩ(x)

)λ,

whereC, λ > 0 depend only onAH andC3 but not onx nor onh. Thus we have a weak estimate

h(x)h(yi)

≤ C

(δΩ(yi)δΩ(x)

)λ. (18)

Let N0 be as in Lemma4.3andR0 be as in the proof of Lemma4.3. We recall again the standingassumption thatX is a proper metric space.

14

Proposition 5.1.Letξ ∈ ∂Ω, 0 < R< R0/16, and h be a positive p-quasiminimizer inΩ∩B(ξ, 16R)vanishing p-quasieverywhere on∂Ω ∩ B(ξ, 16R). If h is bounded inΩ ∩ B(ξ,R/2) \ B(ξ,R/16),then for all x∈ Ω ∩ S(ξ,R/4),

h(x) ≤ CN∑

i=1

h(yi),

where y1, . . . , yN is a system of local reference points forξ (N ≤ N0).

Proof. By (18), for all x ∈ Ω ∩ B(ξ,R/2) there is an integeri ∈ 1, . . . ,N such that

h(x)h(yi)

≤ C

(RδΩ(x)

)λ.

Hence, for everyx ∈ Ω ∩ B(ξ,R/2),

h(x) ≤ C

(RδΩ(x)

)λ N∑i=1

h(yi). (19)

Let

u :=1

C∑N

i=1 h(yi)h.

Thenu is non-negative and locally bounded on the bounded open setΩ0 given byΩ0 := B(ξ,R/2)\B(ξ,R/16). Moreover,u is ap-quasisubminimizer inΩ0, because it is ap-quasiminimizer inΩ0∩Ωand vanishes inΩ0 ∩ ∂Ω; see Lemma 3.11 of [BBS1]. By (19),

u(x) ≤(

RδΩ(x)

)λfor x ∈ Ω0 ∩Ω.

Let τ > 0 be as in Lemma4.4. Chooseε > 0 such thatQ − 1 + ε > 1 and apply the elementaryinequality (

log t)Q−1+ε ≤

(Q− 1+ ετ

)Q−1+ε

tτ whenevert ≥ 1

to the quantityt = R/δΩ(x) ≥ 1 wheneverx ∈ Ω ∩Ω0 to obtain the following estimate:

I =∫Ω∩Ω0

(log+ u(x)

)Q−1+ε dµ(x) ≤∫Ω∩Ω0

[λ log+

(RδΩ(x)

)]Q−1+ε

dµ(x)

≤∫Ω∩Ω0

C

(RδΩ(x)

)τdµ(x).

By Lemma4.4and the Ahlfors regularity ofµ,

I ≤ Cλ,Q,ε

∫Ω∩B(ξ,R/2)

(RδΩ(x)

)τdµ(x) ≤ C RQ < ∞. (20)

Therefore (10) of Lemma3.1yields

u(x) ≤ C exp(C I1/εδΩ0(x)−Q/ε

)wheneverx ∈ Ω ∩Ω0.

15

On the other hand, ifx ∈ Ω ∩ S(ξ,R/4), thenδΩ0(x) ≈ R. Hence by (20),

u(x) ≤ C exp(C I1/εR−Q/ε

)≤ C.

Now in view of the definition ofu, we have the desired result.

As a corollary to Proposition5.1 we have the following Carleson estimate. Such an estimatefollows from Proposition5.1and the strong maximum principle (see [KiSh]) for p-quasiminimizers.

Theorem 5.2. Let Ω ⊂ X be a bounded John domain,ξ ∈ ∂Ω, and 0 < R < R0/16 where R0

is as in the proof of Lemma4.4. If h is a positive p-quasiminimizer inΩ ∩ B(ξ, 16R) vanishingp-quasieverywhere on∂Ω ∩ B(ξ, 16R) and bounded inΩ ∩ B(ξ,R/2), then

h(x) ≤ CN∑

i=1

h(yi) for every x∈ Ω ∩ B(ξ,R/4),

where y1, . . . , yN ∈ Ω ∩ S(ξ,R) are a system of local reference points forξ. The constant C> 1 isindependent of x,ξ, R, and h.

Corollary 5.3. Let Ω ⊂ X be a uniform domain,ξ ∈ ∂Ω, and 0 < R < R0/16 where R0 isas in the proof of Lemma4.4. If h is a positive p-quasiminimizer inΩ ∩ B(ξ, 16R) vanishingp-quasieverywhere on∂Ω ∩ B(ξ, 16R) and bounded inΩ ∩ B(ξ,R/2), then

h(x) ≤ Ch(y) for every x∈ Ω ∩ B(ξ,R/4),

where y∈ Ω ∩ S(ξ,R) is such thatδΩ(y) ≥ R/C. The constant C> 1 is independent of x,ξ, R, andh.

Remark 5.4. In light of Remark2.6, the above theorem and corollary hold ifh is p-harmonic.

6 The conformal Martin boundary of a bounded John domainin X

In this section we assume thatX is of Q-bounded geometry and apply the results of the previoussections withp = Q. Given a Cheeger derivative structure onX, the corresponding conformalMartin boundary has been defined in [HST] (see Definition2.9). In the following, we letλ > 0 beas in (18).

Theorem 6.1. LetΩ ⊂ X be a bounded John domain such that CapQ(X \ Ω) > 0. Let M be aconformal Martin kernel forΩ with fundamental sequence(yk). Then(yk) converges to a pointξM ∈ ∂Ω and the following statements hold:

(i) M is bounded in a neighborhood ofξ for eachξ ∈ ∂Ω \ ξM;

(ii) there is a constant C≥ 1 such that M(x) ≤ C d(x, ξM)−λ for all x ∈ Ω;

(iii) M vanishes continuously Q-quasieverywhere on∂Ω;

16

(iv) there is a sequence(xn) in Ω converging toξM such thatlimn M(xn) = ∞, and hence thepoint ξM is uniquely determined from M but does not depend on the fundamental sequencethat gave rise to it.

A crucial result needed to prove (iv) of the above theorem is Corollary 6.2 of [BBS2]. Observethat Corollary 6.2 of [BBS2] is invalid if CapQ(X\Ω) = 0. The proof of this result uses the fact thatsolutions top-Dirichlet problems with boundary data belonging toN1,Q(X) satisfy a comparisontheorem; see [Sh2] for a proof of this fact under the assumption that CapQ(X \Ω) > 0. However, ifCapQ(X \Ω) = 0 then this comparison theorem obviously fails and as a consequence Corollary 6.2of [BBS2] also fails to hold.

Let U be an open subset ofX. Recall thatHUQ f stands for the solution to theQ-Dirichlet

problem on the open setU with boundary dataf ∈ N1,Q(X). Given a continuous functionf :∂U → R, the authors of [BBS2] construct a Perron solution inU with boundary dataf (in fact,[BBS2] also shows that continuous functions are resolutive); letPU

Q f denote such a solution. Ifsuch a functionf also happens to belong toN1,Q(X), thenPU

Q f = HUQ f . We say thatξ ∈ ∂U is aQ-

regular boundary point forU if lim U∋x→ξ PUQ f (x) = f (ξ) wheneverf is a bounded and continuous

function on∂U. To prove Theorem6.1we need the following lemma.

Lemma 6.2. LetΩ ⊂ X be a bounded domain such that CapQ(X \ Ω) > 0. Then Q-quasi everypoint ξ ∈ ∂Ω is a Q-regular boundary point forΩ ∩ B(ξ, r) whenever r> 0.

Proof. Let (rn) be an enumeration of all positive rational numbers. Moreover, let (xk) be a sequenceof points inΩ that is dense inΩ. By Theorem 3.9 of [BBS1], the setJk,n of all points inB(xk, rn)∩∂Ω that areQ-irregular points forΩ∩B(xk, rn) is of zeroQ-capacity. Hence the setJ :=

∪k,n Jk,n is a

zeroQ-capacity subset of∂Ω. Let ξ ∈ ∂Ω\ J. We will demonstrate thatξ satisfies the requirementsof the above lemma.

Supposeξ does not satisfy the requirements of the above lemma; that is, suppose there isr < diam(Ω) such thatξ is not aQ-regular boundary point forΩ ∩ B(ξ, r). Then, following thenotation set up in the proof of Theorem 3.9, [BBS1], we can find a ballBl,m centered at somepoint in ∂Ω and radiusρl,m such thatξ ∈ Bl,m, 2Bl,m ⊂ B(ξ, r), and a Lipschitz functionφl,m that iscompactly supported in 2Bl,m and takes on the value of 1 inBl,m such that

lim infΩ∩B(ξ,r)∋y→ξ

HΩ∩B(ξ,r)Q φl,m(y) < 1.

On the other hand, as 2Bl,m ⊂ B(ξ, r) and ξ ∈ Bl,m, we can findk,n such thatξ ∈ B(xk, rn) ⊂B(xk, rn) ⊂ B(ξ, r). As ξ < Jk,n, we know thatξ is a Q-regular boundary point of the open setΩ ∩ B(xk, rn). Hence we have by the comparison theorem that

lim infΩ∩B(ξ,r)∋y→ξ

HΩ∩B(ξ,r)Q φl,m(y) ≥ lim inf

Ω∩B(xk,rn)∋y→ξHΩ∩B(xk,rn)

Q φl,m(y) = 1,

contradicting the above inequality. Hence suchr does not exist, and consequentlyξ satisfies thelemma.

Proof of Theorem6.1. Let (yk) be a fundamental sequence inΩ giving rise to the kernelM; seeDefinition 2.9. We will show that there is a pointξM ∈ ∂Ω such that limk yk = ξM. Since (yk) isa fundamental sequence, it has no accumulation point inΩ. SinceΩ ⊂ F is compact, there is a

17

pointξM ∈ ∂Ω and a subsequence (ykn)n such that limn ykn = ξM. We will demonstrate that for everyξ ∈ ∂Ω \ ξM the functionM is bounded in a neighborhood ofξ and thatM satisfies (iii ) and (iv)of the theorem. As a consequence, this observation concludes that limk yk = ξM. Moreover, if (wn)is another fundamental sequence giving rise toM, then limn wn = ξM.

Let us begin with the proof of (i) of the theorem. Fixξ ∈ ∂Ω \ ξM. For ease of notation let uscall the subsequence of (yk) that converges toξM also as (yk). Then there existsr = 4rξ > 0 suchthat for allk ∈ N, yk < Ω∩ B(ξ, 16(q+ 1)r/c) and 16(q+ 1)r/c < R0, whereR0 is as in the proof ofLemma4.3. Let Mk be the function given byMk(x) = g(x, yk)/g(x0, yk). ThenMk is positive andQ-harmonic inΩ ∩ B(ξ, 16(q+ 1)r/c), and vanishesQ-quasieverywhere on∂Ω. Moreover,Mk isbounded onΩ ∩ B(ξ, r), and therefore by Theorem5.2, for x ∈ Ω ∩ B(ξ, r/2) we have

Mk(x) ≤ CN∑

i=1

Mk(yi),

wherey1, . . . , yN ∈ Ω∩S(ξ, r/2) is a system of local reference points forξ. Note thatδΩ(yi) ≥ r/A.Thus we have

kΩ(yi , x0) ≤Acr

d(yi , x0) ≤Acr

diam(Ω),

and asMk(x0) = 1, by Lemma4.6 we haveMk(yi) ≤ exp(AH

Acrdiam(Ω)

), and consequently for

x ∈ Ω ∩ B(ξ, r/4),

Mk(x) ≤N∑

i=1

exp(AH

Acr

diam(Ω))≤ N0 exp

(AH

Acr

diam(Ω))=: Cξ.

SinceM = limk Mk, it follows that M is bounded in a neighborhood ofξ for eachξ ∈ ∂Ω \ ξM.Thus (i) in the theorem is satisfied.

Let us next prove (iii ) of the theorem. Since by assumption CapQ(X \ Ω) > 0, by the Poincareinequality we see that CapQ(∂Ω) is positive. By Lemma6.2 we may assume thatξ ∈ ∂Ω isQ-regular forΩ ∩ B(ξ, r) for every r > 0. We can easily see that wheneverρ > 0 we haveCapQ(B(ξ, ρ) \ Ω) > 0. Let r = rξ/4. We have by the above argument that the familyMkk is auniformly bounded family ofQ-harmonic functions onΩ∩B(ξ, rξ); Mk ≤ Cξ. Let f be a compactlysupported Lipschitz continuous function onX such thatf = Cξ on S(ξ, r) and f = 0 onB(ξ, r/2).Then we easily see that

Mk(y) ≤ f (y) for everyy ∈ ∂(Ω ∩ B(ξ, r)),

sinceMk = 0 onX \Ω by the construction ofMk in [HoSh]. Note that CapQ(B(ξ,2r) \Ω) > 0 andthat bothf andMk are in the classN1,Q(B(ξ, 2r)) and areQ-harmonic inB(ξ, r)∩Ω for sufficientlylargek. Therefore by the regular comparison theorem (see [Sh2]) we haveMk ≤ HΩ∩B(ξ,r)

Q f onΩ ∩ B(ξ, r). Sinceξ is aQ-regular boundary point forΩ ∩ B(ξ, r) and f is a continuous boundarydata, it follows that for everyε > 0 there can be foundρε > 0 such that 0< HΩ∩B(ξ,r)

Q f < ε onΩ ∩ B(ξ, ρε). Hence 0< Mk(x) < ε onΩ ∩ B(ξ, ρε) for everyk. Thus, asM is the pointwise (andlocally uniform) limit of Mk in Ω, we see thatM ≤ ε onΩ ∩ B(ξ, ρε). This means thatM tends tozero continuouslyQ-quasieverywhere in∂Ω. This proves (iii ) of the theorem.

Now we prove (iv) of the theorem. Suppose that there is no sequence (xn) in Ω converging toξM for which limn M(xn) = ∞. ThenM is also bounded in a neighborhood ofξM, and consequently

18

by (i) of the theorem (proven above),M is a positiveQ-harmonic function inΩ that is boundedonΩ and in addition vanishesQ-quasieverywhere in∂Ω. It then follows from Corollary 6.2 of[BBS2] that M is identically zero onΩ, contradicting the fact thatM(x0) = 1. Thus (iv) of theabove theorem is also true.

It now only remains to prove (ii ). We have already shown that wheneverR> 0 the functionMis bounded inΩ∩B(ξM,R/2)\B(ξM,R/16) and vanishes on∂Ω\ξM. Therefore by Proposition5.1,if 0 < R< R0/32, thenM satisfies

M(x) ≤ CN∑

i=1

M(yi) for x ∈ S(ξM,R) ∩Ω,

wherey1, . . . , yN ∈ Ω ∩ S(ξM,2R) is a system of local reference points of orderN for ξM. By thecomparison theorem we have

M(x) ≤ CN∑

i=1

M(yi)

wheneverx ∈ Ω \ B(ξM,R). Now an application of (18) to each of the reference pointsyi togetherwith the estimateδΩ(yi) ≥ R/C gives

M(yi) ≤ M(x0)

(δΩ(x0)

R

)λ=

(δΩ(x0)

R

)λ= CR−λ.

HenceM(x) ≤ C N R−λ ≤ C N0 d(x, ξM)−λ for x ∈ Ω ∩ B(ξM,2R) \ B(ξM,R),

whenever 0< R< R0/32. The desired result now follows from the fact thatΩ is bounded.This completes the proof of Theorem6.1.

To obtain Theorems5.2 and6.1 as well as other results in this and the previous sections, itsuffices to only know thatQ-quasievery boundary point ofΩ is a “John point”. More specifically,a pointξ ∈ ∂Ω is said to be aJohn pointif there is a radiusRξ > 0, a pointxξ ∈ Ω, and a constantCξ > 0 such that for everyx ∈ B(ξ,Rξ) ∩ Ω there is a compact rectifiable curveγ connectingxto xξ with γ ⊂ Ω ∩ B(ξ,CξRξ) such that for everyy ∈ γ we haveδΩ(y) ≥ ℓ(γx,y)/Cξ, see [Sh3].Examples of domains satisfying the above weak condition include Euclidean domains obtained bypasting outward pointing cusps to a ball.

7 Compactness ofX

The proof of the results discussed in this note required that the closure ofΩ be compact. In thissection we will demonstrate that instead of merely assuming the closure ofΩ be compact we mayassume the stronger condition thatX is a complete metric space. Since a metric space supporting adoubling measure is totally bounded, ifX is complete then it is proper as well and hence boundeddomains inX are relatively compact.

If X is not a complete metric space, letX denote the completion ofX, d the extension ofd andlet µ be the extension ofµ to X: µ(A) = µ(A∩ X) wheneverA ⊂ X.

19

Proposition 7.1. If (X,d, µ) is a metric measure space with doubling measureµ and supporting a(1, p)-Poincare inequality, then so is(X, d, µ). In this case, N1,p(X) = N1,p(X). In addition, ifµ isAhlfors Q-regular, then so isµ.

Proof. We first prove the doubling property ofµ. Let B(x, r) be a ball inX. If x ∈ X, then thedoubling property ofµ itself guarantees the doubling inequality forµ: µ(B(x,2r)) = µ(B(x,2r) ∩X) ≤ Cdµ(B(x, r) ∩ X) = Cdµ(B(x, r)). Supposex ∈ X \ X. Then asX is the completion ofX, thereis a pointx′ ∈ X such thatd(x, x′) < r/8. Thus by the doubling property ofµ,

µ(B(x, 2r)) ≤ µ(B(x′,3r)) = µ(B(x′,3r) ∩ X) ≤ C2dµ(B(x′,3r/8)∩ X)

≤ C2dµ(B(x, r) ∩ X) = C2

dµ(B(x, r)).

That is, µ is doubling with doubling constantC3d. A similar argument demonstrates that ifµ is

Ahlfors Q-regular then so isµ.Now we demonstrate that ifµ is doubling and supports a (1, p)-Poincare inequality thenµ also

supports a (1, p)-Poincare inequality. To this end, letu ∈ N1,p(X) andg be an upper gradient ofuin X. Note thatg is also an upper gradient ofu in X asX ⊂ X. SupposeB(x, r) ⊂ X. If x ∈ X, thenby the definition ofµ and by the fact thatµ itself supports a (1, p)-Poincare inequality we have thePoincare inequality with respect toX. If x ∈ X \ X, then takingx′ ∈ X such thatd(x, x0) < r/2, weobtain

infc∈R

∫B(x,r)

|u− c| dµ ≤ infc∈R

∫B(x′,2r)

|u− c| dµ ≤∫

B(x′,2r)∩X

|u− uB(x′,2r)∩X| dµ

≤ Cp 4r∫

B(x′,2κr)∩X

gp dµ ≤ Cr∫

B(x,3κr)∩X

gp dµ = Cr∫

B(x,3κr)

gp dµ.

Hence the pairu,g satisfies a (1, p)-Poincare inequality onX.Finally, we prove thatN1,p(X)|X = N1,p(X). Observe thatN1,p(X)|X ⊂ N1,p(X). On the other

hand, asX supports a (1, p)-Poincare inequality, we know from [Sh1] that Lipschitz functions aredense inN1,p(X). We may extend a Lipschitz functionu ∈ N1,p(X) to X, for example via a McShaneextension. Since the minimalp-weak upper gradientgu of such a function has the property thatgu(x) ≈ Lip u(x) for µ-a.e. x ∈ X (see for example [Ch]) and hence forµ-a.e. x ∈ X, we seethatu ∈ N1,p(X) with theN1,p(X)–norm ofu equaling theN1,p(X)–norm ofu. Now the density ofLipschitz functions inN1,p(X) guarantees the coincidence of the two function spaces.

The above result demonstrates that the requirement ofX being proper is not a stringent re-quirement as it appears to be, especially since we do not assume much aboutX \ Ω apart from thecondition that CapQ(X \ Ω) > 0. The following proposition further strengthens this claim. Note

that for every domainΩ ⊂ X there is a domainΩ0 ⊂ X such thatΩ = Ω0 ∩ X. LetΩ0 denote thelargest such domain inX.

Proposition 7.2. In the situation of Proposition7.1, if CapQ(X \ X) = 0 then∂cMΩ = (Ω0 \ Ω) ∪∂cMΩ0.

Proof. In this proof, ballsB(x, r) denote balls inX; B(x, r) = y ∈ X : d(y, x) < r. By B(x, r) wethen denote the closure ofB(x, r) in X.

20

Let (r i) be a sequence of positive real numbers such that limi r i = 0. If y ∈ Ω ⊂ Ω0, theconstruction of aQ-singular function onΩ with singularity aty is through a sequence of functionsui ∈ N1,Q

0 (Ω) that areQ-energy minimizers inΩ \ B(y, r i), take on the value 1 inB(y, r i), and 0≤ui ≤ 1 onX; see [HoSh]. By a Q-energy minimizer we mean that for everyϕ ∈ N1,Q

0 (Ω \ B(y, r i)),∫Ω\B(y,r i )

|dui |Q dµ ≤∫Ω\B(y,r i )

|d(ui + ϕ)|Q dµ.

Since by Proposition7.1 we haveN1,Q0 (Ω) ⊂ N1,Q(X) = N1,Q(X), we see thatui is extendable

to Ω0 so thatui ∈ N1,Q0 (Ω0). We claim thatui is Q-harmonic inΩ0 \ B(y, r i). Indeed, ifφ ∈

N1,Q0 (Ω0 \ B(y, r i)), thenφ

∣∣∣X∈ N1,Q

0 (Ω), and hence∫Ω0\B(y,r i )

|dui |Q dµ =∫Ω\B(y,r i )

|dui |Q dµ

≤∫Ω\B(y,r i )

|d(ui + φ)|Q dµ =∫Ω0\B(y,r i )

|d(ui + φ)|Q dµ.

Therefore the singular functiong(·, y) onΩ can be extended to be a singular function on all ofΩ0

with singularity aty.Now let M ∈ ∂cMΩ and let (yn) be an associated fundamental sequence inΩ. As this sequence

can have no accumulation point inΩ, and asΩ0 is compact, we see that this sequence has accumu-lation points in (Ω0\Ω)∪∂cMΩ0. If it has an accumulation point inΩ0\Ω, thenM is the normalizedsingular function onΩ0 with singularity at that accumulation point; hence the sequence (yn) mustconverge to this unique accumulation point. Otherwise all of the accumulation points of the se-quence lie in the set∂Ω0, and henceM ∈ ∂cMΩ0. It therefore follows that∂cMΩ ⊂ (Ω0\Ω)∪∂cMΩ0.

To get equality note that every point inΩ0 \ Ω corresponds to a unique Martin kernel functionin ∂cMΩ. Moreover, the restriction of every function in∂cMΩ0 toΩ lies in∂cMΩ. To see this, notethat every pointy0 ∈ Ω0 is a limit of a sequence of points fromΩ; hence every singular functionin Ω0 with singularity aty can be approximated to any desired accuracy by a singular function inΩ with a singularity inΩ; this is because by the above discussion singular functions inΩ withsingularity inΩ are extendable to be a singular function inΩ0 with singularity inΩ, and the limitof such a sequence of singular functions is a singular function inΩ0 with singularity aty, and onlyone such singular function exists. Now a diagonalization argument yields that∂cMΩ0 ⊂ ∂cMΩ.

References

[Ai1] H. Aikawa. Integrability of superharmonic functions in a John domain,Proc. Amer.Math. Soc.128(2000) 195–201.

[Ai2] H. Aikawa.Boundary Harnack principle and Martin boundary for a uniform domain,J. Math. Soc. Japan53 (2001) 119–145.

[AHL] H. Aikawa, K. Hirata, and T. Lundh.Martin boundary points of John domains andunions of convex sets,preprint.

21

[AO] H. Aikawa and M. Ohtsuka.Extremal length of vector measures,Ann. Acad. Sci. Fenn.Math.24 (1999) 61–88.

[Be] M. Benedicks,Positive harmonic functions vanishing on the boundary of certain do-mains inRn, Ark. Mat. 18 (1980), no. 1, 53–72.

[BBS1] A. Bjorn, J. Bjorn, and N. Shanmugalingam.The Dirichlet problem for p-harmonicfunctions on metric measure spaces,J. Reine Angew. Math.(Crelle)556 (2003) 173–203.

[BBS2] A. Bjorn, J. Bjorn, and N. Shanmugalingam.The Perron method for p-harmonic func-tions in metric spaces,J. Differential Equations195(2003) 398–429.

[BHK] M. Bonk, J. Heinonen, and P. Koskela.Uniformizing Gromov hyperbolic spaces,Asterisque270(2001) .

[Ca] L. Carleson,On the existence of boundary values for harmonic functions in severalvariables, Ark. Mat. 4 (1962), 393–399.

[Ch] J. Cheeger.Differentiability of Lipschitz functions on metric measure spaces,Geom.Funct. Anal.9 (1999) 428–517.

[Chv] N. Chevallier,Frontiere de Martin d’un domaine deRn dont le bord est inclus dansune hypersurface lipschitzienne, Ark. Mat. 27 (1989), no. 1, 29–48.

[D] Y. Domar.On the existence of a largest subharmonic minorant of a given function,Ark. Mat. 3 (1957) 429–440.

[FHK] B. Franchi, P. Hajłasz, and P. Koskela.Definitions of Sobolev classes on metric spaces,Ann. Inst. Fourier (Grenoble)49 (1999) 1903–1924.

[HaK1] P. Hajłasz and P. Koskela.Isoperimetric inequalities and imbedding theorems in irreg-ular domains,J. London Math. Soc. (2)58 (1998) 425–450.

[HaK2] P. Hajłasz and P. Koskela.Sobolev met Poincare,Mem. Amer. Math. Soc.145(2000) .

[He] J. Heinonen.Lectures on Analysis on Metric Spaces,Springer–Verlag, New York,(2001).

[HKM] J. Heinonen, T. Kilpelainen, and O. Martio.Nonlinear Potential Theory of DegenerateElliptic Equations,Oxford Univ. Press, New York, (1993).

[HeK1] J. Heinonen and P. Koskela.Quasiconformal maps in metric spaces with controlledgeometry,Acta Math.181(1998), 1–61.

[HeK2] J. Heinonen and P. Koskela.A note on Lipschitz functions, upper gradients, and thePoincare inequality,New Zealand J. Math.28 (1999) 37–42.

[Ho] I. Holopainen.Nonlinear potential theory and quasiregular mappings on Riemannianmanifolds,Ann. Acad. Sci. Fenn. Ser. A I Math. Diss.74 (1990) 1–45.

22

[HoSh] I. Holopainen and N. Shanmugalingam.Singular functions on metric measure spaces,Collect. Math.53 (2002) 313–332.

[HST] I. Holopainen, N. Shanmugalingam, and J. Tyson.On the conformalMartin boundary of domains in metric spaces,Papers on analysis, Rep.Univ. Jyvaskyla Dep. Math. Stat., 83 (2001) 147–168, also found inhttp://www.math.jyu.fi/research/report83.html.

[JK] D. S. Jerison and C. E. Kenig,Boundary behavior of harmonic functions in nontan-gentially accessible domains, Adv. in Math.46 (1982), no. 1, 80–147.

[KZ] S. Keith and X. Zhong.The Poincare inequality is an open ended condition,preprint,available athttp://wwwmaths.anu.edu.au/ keith/selfhub.pdf.

[KiMa0] J. Kinnunen and O. Martio.The Sobolev capacity on metric spaces,Ann. Acad. Sci.Fenn. Math.21 (1996) 367–382.

[KiMa1] J. Kinnunen and O. Martio.Nonlinear potential theory on metric spaces,Illinois J.Math.46 (2002) 857–883.

[KiMa2] J. Kinnunen and O. Martio.Potential theory of quasiminimizers,Ann. Acad. Sci. Fenn.Math.28 (2003) 459–490.

[KiSh] J. Kinnunen and N. Shanmugalingam.Regularity of quasi-minimizers on metricspaces,Manuscripta Math.105(2001) 401–423.

[KoMc] P. Koskela, P. MacManus.Quasiconformal mappings and Sobolev spaces,Studia Math.131(1998) 1–17.

[L] J. Lewis.Uniformly fat sets,Trans. Amer. Math. Soc.308(1988) 177–196.

[Mi] P. Mikkonen.On the Wolff potential and quasilinear elliptic equations involving mea-sures,Ann. Acad. Sci. Fenn. Math. Diss.104(1996).

[Sh1] N. Shanmugalingam.Newtonian spaces: an extension of Sobolev spaces to metric mea-sure spaces,Rev. Mat. Iberoamericana16 (2000) 243–279.

[Sh2] N. Shanmugalingam.Harmonic functions on metric spaces,Illinois J. Math.45 (2001)1021–1050.

[Sh3] N. Shanmugalingam.Singular behavior of conformal Martin kernels and non-tangential limits of conformal mappings,Ann. Acad. Sci. Fenn. Math.29 (2004) 195–210.

Address:H.A.: Department of Mathematics, Shimane University, Matsue 690–8504, Japan.E-mail: haikawa@math.shimane-u.ac.jpN.S.: Department of Mathematical Sciences, P.O.Box 210025, University of Cincinnati, Cincin-nati, OH 45221–0025, U.S.A.E-mail: nages@math.uc.edu

23