ON QUASIEXTREMAL DISTANCE DOMAINS IN METRIC SPACES ANCA ANDREI The purpose of this note is to give some properties of a quasiextremal distance domain in a

ANCA ANDREI

The purpose of this note is to give some properties of a quasiextremal distance domain in a Q-Ahlfors regular Loewner space. In the first part of this note, we recall certain properties about domains and balls in a geodesic space that have been proved in [2]. Next, we shall introduce the notion of weakly quasiconformal accessibility. Gehring and Martio have introduced the class of quasiextremal dis- tance domains in the euclidean case. They proved that a quasiextremal distance domain is linearly locally connected and hence, locally connected on the bound- ary. We shall prove that, in a locally path connected and Q-Ahlfors regular Loewner space, a bounded quasiextremal distance domain is locally connected and weakly quasiconformal accessible on the boundary. Finally, we deal with the extension theorem to boundary for quasiconformal mappings on domains in Q-Ahlfors regular Loewner spaces, when one of the domains is a quasiextremal distance domain.

AMS 2010 Subject Classification: 30C65, 30C75.

Key words: quasiconformal mapping,Q-Ahlfors regular Loewner space, locally path connected, weakly quasiconformal accessibility, quasiextremal distance domain.

This note is a continuation of my papers [1, 2] and consequently, the notations are the same like in [1, 2].

At the begining, we recall some definitions and results which we shall use in this paper. Next we give certains properties of a quasiextremal distance domain in aQ-Ahlfors regular Loewner space. Finally, we deal with boundary extension theorems for quasiconformal mappings when one of the domains is a quasiextremal distance domain.

Throughout the paper, we shall consider only metric spaces. LetX be a metric space. The space X is said to be path connected (or arc connected) if for any two points xand y inX there exists a continuous functionγ from the unit interval [0,1] toX with γ(0) = x and γ(1) = y. This function is called a path from x to y. We say that x and y are the endpoints of γ and that γ joins (or connects) the points x and y. An arc domain Din X is an open arc connected set inX. By a continuum, we mean a compact connected set which contains at least two points.

MATH. REPORTS15(65),4(2013), 319–329

A path is rectifiable if its length is a finite number. A metric space is said to be rectifiably connected if any two points can be joined by a rectifiable path. By a curve in a metric space X we mean either a path γ or its image.

We usually abuse notation and callγ both the path and its image.

Note that every path connected metric spaces is connected and the image of a path is always a path connected compact space.

A metric spaceX is said to be locally (path) connected if for eachx∈X and each neighbourhood U of x there exists a (arc) connected neighbourhood V of x such that V ⊂U. It is known that X is locally connected if and only if the following property holds: for each open subsetS of X , each component of S is an open set ([3], Theorem 5.16).

Let (X, d) be a metric space. We denote byB(x, r) ={y∈X, d(x, y)<

r} the open ball of center x in X and radius r < diamX and its closure B(x, r).The closed ball inX centered at the point x and with radiusr is the set B[x, r] = {y ∈ X, d(x, y) ≤ r}. It is known that B(x, r) ⊂ B[x, r] and B(x, r)⊂IntB[x, r].Also, we know that open balls of a metric space are open sets and closed balls are closed sets. In this context, we recall the next theorem.

Remark 1 ([11], Theorem 5.1.7).Suppose that (X, d) is a metric space, x∈X and r >0.Then:

(i) ∂B(x, r)⊂ {y, d(x, y) =r}; (ii) ∂B[x, r]⊂ {y, d(x, y) =r}.

Next, we recall some results which we shall use in this note.

Proposition1 ([2], Theorem 2.1). IfXis rectifiably connected and locally path connected space and D is a domain inX, then D is an arc domain.

Lemma 1 ([14], 11.26). LetE be a connected subset of a topological space X. IfA⊂X and neither E∩A nor E∩(X\A)is empty, then E∩∂A6=∅. Definition 1 ([10], Definitions 2.2.1, 2.4.1).Ageodesic path(or, simply, a geodesic) in a metric space X is a path γ which connects two points in X and the length ofγ is equal to the distance between the points. A metric space is called geodesic space if every pair of distinct points can be connected by a geodesic.

Also, we recall some results about domains and balls in geodesic metric spaces.

Proposition 2 ([2], Theorem 2.2). If X is a geodesic metric space then B(x, r) =B[x, r], where x∈X and r < diamX.

Remark 2. A ball of a connected metric space doesn’t need to be con- nected, but it is known that in a compact metric space X with the property

“the closure of any open ballB(x, r) in X is the closed ballB[x, r]”, then any open or closed ball is connected. Therefore, if (X, d) is a compact geodesic met- ric space, by Proposition 2, X has previously mentioned property and hence, any ball is connected.

Proposition 3 ([2], Proposition 2.1). If (X, d) is a geodesic space then any ball B(a, r) with center a in X and radius r < diamX, is arc connected.

Moreover, any geodesic which connects two points in B(a, r) has length <2r and it lies in B(a,2r).

Proposition 4 ([2], Corollary 2.1). If (X, d) is a geodesic metric space then X is locally path connected, and consequently, locally connected.

Remark 3. Using Proposition 3 and Remark 2.3 [2] it follows that, if (X, d) is a geodesic space then any ball in X is arc connected.

Let us consider (X, d, µ) a metric measure space, D a domain in X and b a boundary point ofD.

Definition 2. Dislocally (arc) connectedatbifbhas arbitrarily small neighbourhoods U such that U ∩D is (arc) connected. This means that for each neighbourhood V of b there exists a neighbourhood U of b,U ⊂V such that U∩Dis (arc) connected.

Let (X, d, µ) be a metric measure space and let Γ be a family of (noncon- stant) curves inX. We say that a Borel function ρ:X→[0,∞] is admissible

for Γ if Z

γ

ρds≥1

for all locally rectifiable pathsγ in Γ. The set of all admissible function for Γ is denoted byF(Γ).

Definition 3. The (conformal)p-modulusof Γ, 0< p <∞, is defined as:

M_p(Γ) = inf

ρ∈F(Γ)

Z

X

ρ^pdµ.

Note in particular, that the modulus of the collection of all non-locally rectifiable paths is zero.

Definition 4.A metric measure space (X, d, µ) is said to be Q-Ahlfors regular,Q >0, if there exists a positive constant C <∞ such that

r^Q

C ≤µ(B_r)≤Cr^Q for every ballB_r inX with radius r < diamX.

Note that if the above relation holds then X has Hausdorff dimension equal toQ.

IfE, F, Dare subsets ofXwithE⊂D,F ⊂D, we denote by ∆(E, F;D) the family of all paths which joinE andF inD. The notation ∆(E, F) means the family of all paths which joinE and F inX.

Definition 5.Suppose that (X, d, µ) is a metric measure space of Haus- dorff dimension Q, (Q >0) and rectifiably connected. We call X a Loewner space (or a Q-Loewner space) if there is a function ϕ : (0,∞) → (0,∞) so that

MQ(∆(E, F))≥ϕ(t)

whenever E and F are two disjoint, nondegenerate continua inX and t≥δ(E, F) = dist(E, F)

min{diamE, diamF}.

Remark 4. By Theorem 3.6 [5], it follows that ifX is aQ-Ahlfors regular Loewner space, Q > 1, then there exists a decreasing homeomorphism ψ : (0,∞) → (0,∞) such thatM_Q(∆(E, F))≥ψ(t). Moreover, one can selectψ so as to satisfy ψ(t) ≈log¹_t,for all sufficiently smallt, and ψ(t)≈(logt)^1−Q for all sufficiently large t.

Remark 5. By combining Theorem 3.6 [5] and Theorem 3.13 [5], we can deduce that a Q-Ahlfors regular Loewner space (Q > 1) is linearly locally connected and quasiconvex.

Definition 6.Suppose that (X, d, µ) is a metric measure space of Haus- dorff dimension Q, (Q > 0).A domain D in X is called k-quasiextremal distance domain, k≥1, if the following condition holds:

M_Q(∆(E, F))≤k·M_Q(∆(E, F;D))

for any disjoint continuaEandF inD. We abbreviate this situation by saying that Dis k-QED. A domain is QED if it isk-QED for some constants k≥1.

Gehring and Martio [4] introduced the class of quasiextremal distance domains in the euclidean case. They proved that a quasiextremal distance domain is linearly locally connected ([4], 2.11) and hence, locally connected on the boundary. In this note, we shall prove that, in a locally path connected and Q-Ahlfors regular Loewner space, a quasiextremal distance domain is locally connected and weakly quasiconformal accessible on the boundary.

Next, we shall introduce the definition of a weakly quasiconformal acces- sible domain at a boundary point.

Definition 7. Suppose that (X, d, µ) is a metric measure space. A domain D in X is weakly quasiconformal accessible at a boundary point b if for each neighbourhood U of b, there exist a neighbourhood V of b, V ⊂ U, a continuum F inD and a constantδ >0 such that

MQ(∆(E, F;D))≥δ

for all open connected sets E⊂D, withE∩∂U 6=∅and E∩∂V 6=∅. The definition of a domain weakly quasiconformal accessible at a bound- ary point differs from strictly quasiconformal accessibility at a boundary point ([1], Definition 10) since instead of connected sets we have considered open connected sets.

Obviously, strictly quasiconformal accessibility implies weakly quasicon- formal accessibility.

Next, we give some results about quasiextremal distance domains. We consider only rectifiable connected spaces.

Theorem 1. Suppose that X is a Q-Ahlfors regular Loewner space, Q > 1, and locally path connected. If D ⊂ X is a bounded QED domain then Dis locally arc connected on the boundary.

Proof. Suppose that D is a k-QED domain with k ≥ 1. Since X is rectifiably connected and locally path connected, by Proposition 1, it follows thatD is arc connected. For the first part of the proof, we use arguments like in Lemma 13.18 [8]. Consider a boundary point bof D.Let us assume that D is not locally arc connected atb. Denote by d₀ = sup

x∈D

d(x, b).

SinceDis not locally arc connected atbit follows that there isr0 ∈(0, d0) such that µ(D∩B(b, r₀))<∞ and for every neighbourhood V ⊂B(b, r₀) of the point b, at least one of the following conditions holds:

(*) V ∩D has at least two arc connected components C₁ and C₂ such that b∈C₁∩C₂;

(**) V ∩D has infinitely many arc connected components C₁, ..., C_m, ...

such thatb= lim

m→∞b_m for someb_m ∈C_m, andb /∈C_m for allm= 1,2, .... Note that Cm∩∂V 6=∅for all m= 1,2, ...in view of the arc connectedness ofD.

In particular, either (*) or (**) holds for the neighbourhoodV =B(b, r0). Letr1∈(0, r0). We obtain:

(1) MQ ∆ C_i^∗, C_j^∗;D

≤ µ(D∩B(b, r0)) [2 (r₀−r₁)]^Q <∞, whereC_i^∗ =C_i∩B(b, r₁) , C_j^∗ =C_j∩B(b, r₁) , for all i6=j.

We shall prove the last assertion. One of the admissible functions for the family Γij of all rectifiable curves in ∆

C_i^∗, C_j^∗;D

is ρ(x) = _2(r¹

0−r1) for all x ∈ B(b, r0)\B(b, r1) and ρ(x) = 0 for all x ∈ X\

B(b, r0)\B(b, r1)

, since components C_i and C_j cannot be connected by a path in V ∩D, where V =B(b, r₀), and every path connectingC_i^∗andC_j^∗inDat least twice intersect the ring B(b, r₀)\B(b, r₁).Using Proposition 13.13 [8], we obtain (1).

We denote δ = ^µ(D∩B(b,r_[2(r ⁰⁾⁾

0−r1)]^Q and use (1) we get:

(2) M_Q ∆ C_i^∗, C_j^∗;D

≤δ, for all i6=j.

Since D is k-QED and X is a Q-Ahlfors regular Loewner space, there exists a decreasing homeomorphism ϕ: (0,∞)→(0,∞) such that

(3) k·M_Q ∆ C_i^∗, C_j^∗;D

≥M_Q ∆ C_i^∗, C_j^∗

≥ϕ δ C_i^∗, C_j^∗ , whereδ

C_i^∗, C_j^∗

= ^dist(^Ci^∗,C_j^∗)

min{diamC_i^∗,diamC_j^∗}.

By Remark 4, we can choose r ∈ (0, r₁) such that ϕ δ

C_i^∗

0, C_j^∗

0

≥ k(δ+ 1) for some pairi₀ andj₀,i₀ 6=j₀, sincedist

C_i^∗

0, C_j^∗

0

→0 whenr→0 and in the correspondingC_i^∗₀ andC_j^∗₀ there exists at least one curve intersecting

∂B(b, r₁) and∂B(b, r). Hence, we get the relation:

(4) M_Q ∆ C_i^∗₀, C_j^∗₀;D

≥δ+ 1, which contradicts the relation (2).

Therefore, the assumption on the absence of the arc connectedness of D at the point b was not true.

Theorem 2. Suppose that X is a Q-Ahlfors regular Loewner space, Q > 1, and locally path connected. If D ⊂ X is a bounded QED domain then Dis weakly quasiconformal accessible at each boundary point.

Proof. Since X is Q-Ahlfors regular Loewner space, it follows that there exists a decreasing functionϕ: (0,∞)→(0,∞) such that

(*) MQ(∆(E, F))≥ϕ(t)

whenever E and F are disjoint, nondegenerate continua inX satisfying dist(E, F)≤t·min{diamE,diamF}.

Let x be a boundary point of D and let U be a neighbourhood of x.

Hence, there exists a ballB(x, r) of center xand radius r < diamD such that B(x, r)⊂U.We denote S(x, r) ={y∈X, d(x, y) =r}.

We can take two points a ∈ S(x,^2r₃) ∩D and b ∈ S(x,^5r₆) ∩D. By Proposition 1, we have thatDis an arc domain and hence, there exists a curve γ inDwhich connectsaandb.We can pick a subcurveγ⁰ ofγ which connects S(x,^2r₃) and S(x,^5r₆) in B[x,^5r₆]\B(x,^2r₃).Thus,l(γ⁰)≥d(a, b)≥ ^r₆.

We set F =γ, which is a continuum inD, and V =B(x,^r₃)⊂U. Every open connected set E in D which intersects ∂U and ∂V will also intersects S(x,^2r₃) and S(x,^5r₆). By Proposition 1, E is arc connected and hence, we can choose a curve γ₁ in E which connects S(x,^r₂) and S(x,^r₃) and lies in B[x,^r₂]\B(x,^r₃).SetE⁰ =γ1.Note thatE⁰ is a continuum and E⁰∩F =∅.

On the other hand,

min{diamE⁰, diamF} ≥ r 6 that implies

dist(E⁰, F)

min{diamE, diamF} ≤ 6dist(E⁰, F)

r ≤ 6diamD

r . Using (*) and the fact that Dis QED, we obtain

k·M_Q(∆(E, F;D))≥k·M_Q(∆ E⁰, F;D )

≥M_Q(∆(E⁰, F))≥ϕ

6diamD r

>0.

We denote δ = ¹_kϕ ^6diamD_r

and hence, M_Q(∆ (E, F;D))≥δ,

whenever E ⊂D is a open connected set with E∩∂U 6=∅and E∩∂V 6=∅. Thus, we get the desired conclusion.

Remark 6.By the proof of Theorem 2 it follows that, in the hypothesis of Theorem 2,D is arc strictly quasiconformally accessible on the boundary.

Let (X, d, µ) and (Y, d⁰, µ⁰) be two Q-Ahlfors regular metric measure spaces with 1≤Q <∞, and two domainsD⊂X,D⁰⊂Y.

Definition 8. A homeomorphismf :D→D⁰is calledK-(geometrically) quasiconformal,K∈[1,∞) if

MQ(Γ)

K ≤M_Q(f(Γ))≤KM_Q(Γ)

for every family Γ of paths inD. We also say that a homeomorphismf :D→ D⁰ is (geometrically) quasiconformal if f is K-(geometrically) quasiconformal for someK ∈[1,∞),i.e., if the distortion of moduli of path families under the mappingf is bounded.

To simplify the notation, we write quasiconformal mapping instead of geometrically quasiconformal mapping.

Remark 7.Using the results of [5] and [8], it follows that all three defini- tions (metric quasiconformality, quasisymetry, and geometric quasiconformal- ity) are equivalent for homeomorphism between Q-Ahlfors regular Loewner spaces (Q >1).

Recall that if Ω and Ω⁰ are open sets in metric spaces (X, d) and (Y, d⁰), correspondingly, and f : Ω→ Ω⁰ is a homeomorphism, then the cluster set of f at every pointb∈∂Ω,

C(b, f) = n

b⁰ ∈Y :b⁰ = lim

n→∞f(xn), xn→b, xn∈Ω o

, belongs to the boundary of the set Ω⁰.([8], Proposition 13.14)

In euclidean case, Herron and Koskela proved that if f : D → D⁰ is a quasiconformal mapping, where D is locally connected at a boundary point b andD⁰ is QED, thenf has a continuous extension toD∪{b}([7], Theorem 3.3).

We give an analogously theorem for quasiconformal mappings on domains in Q-Ahlfors regular metric measures spaces.

Next, we consider X and Y two Q-Ahlfors regular metric space,Q >1, D a domain in X and D⁰ a domain in Y.

Theorem 3. Assume that f : D → D⁰ is a quasiconformal mapping.

If D is locally connected at a boundary point b, D⁰ is weakly quasiconformal accessible at least one point of C(f, b) and D⁰ is compact, then there exists the limit of f at b.

Proof. Suppose that C(f, b) contains two distinct points b⁰₁, b⁰₂ and that D⁰ is weakly quasiconformal accessible at b⁰₁. Let U be a neighbourhood of b⁰₁ such that b⁰₂ ∈/ U . By the definition of weakly quasiconformal accessibility of D⁰ atb⁰₁, there exist a neighbourhood V of b⁰₁, V ⊂ U, a continuum F ⊂ D⁰ and a positive number δ >0 such that

(1) M(∆(E, F;D⁰))≥δ

whenever E is a subdomain ofD⁰ withE∩∂V 6=∅,E∩∂U 6=∅. Let us denote by d₀ = sup

x∈D

d(x, b) and take r₀ ∈ (0, d₀) and a natural number n₀ such that 0 < _n¹

0 < r₀. Since D is locally connected at b there is a neighbourhood V_n0 of b such that V_n0 ⊂B_n0 = B(b,_n¹

0) and V_n0 ∩D is connected.

Next, we consider a ball Bn1 = B(b,_n¹

1) ⊂Vn0 with n1 > n0, where n1

is a natural integer. Applying the same procedure, we can choose a sequence V_n0, V_n1, ....of neighbourhoods ofbsuch that everyC_ni =V_ni∩Dis connected,

open and B_ni+1 ⊂ V_ni ⊂ B_ni for all i ≥ 0. Note that dist(b, C_ni) → 0 as i → ∞. Set Γ_i = ∆(C_ni, f⁻¹(F);D) and Γ⁰_i = ∆(f(C_ni), F;D⁰). Since f is a homeomorphism it follows that f(Cni) is connected, open and f⁻¹(F) is a continuum.

On the other hand, by the inclusion C(f, b) ⊂ f(V_ni∩D) = f(C_ni) we obtain that b⁰₁, b⁰₂ ∈f(Cni) and using the relation (1) we get

(2) M(Γ⁰_i)≥δ

for any i. In view of quasiconformality of f, there exists a constant K > 0 such that M(Γ⁰_i)≤K·M(Γ_i) and therefore,

(3) M(Γi)≥ δ

K for any i.

Since f⁻¹(F) is a continuum, f⁻¹(F) ⊂D, D is open, b ∈∂D it follows thatb /∈f⁻¹(F) and forilarge,Bni∩f⁻¹(F) =∅and hence,Cni∩f⁻¹(F) =∅.

Fix such an i and denoted by ri = _n¹

i. Every path connecting f⁻¹(F) and C_ni+j intersects∂B_ni and ∂B_ni+j forj≥1. Butr_k→0 ask→ ∞ and hence, 0<2ri+j < ri<∞forj large.

By Lemma 3.14 [5] there exists a constantC₀ (depending only on Qand the constantC byQ-regularity of X) so that

(4) M(Γ_i+j)≤C₀(log r_i

r_j+i)^1−Q. Since (log _r^ri

j+i)^1−Q → 0 asj → ∞, the relation (4) contradicts the rela- tion (3). Consequently, C(f, b) has, at most, one point for that D⁰ is weakly quasiconformal accessible and hence, we get the desired conclusion.

Corollary 1. Suppose that Y is a Q-Loewner space, locally path con- nected and let f : D → D⁰ be a quasiconformal mapping. If D is locally connected at a boundary point b, D⁰ is QED and D⁰ is compact, then there exists the limit of f at b.

Proof. Since Y is a Q-Loewner space, locally path connected, by Theo- rem 2, it follows that D⁰ is weakly quasiconformal accessible on the boundary.

Using Theorem 3, we obtain the desired conclusion.

Corollary 2. Suppose that Y is a Q-Loewner space, locally path con- nected and let f :D→ D⁰ be a quasiconformal mapping. If D is locally con- nected on the boundary,D⁰ is QED and D⁰ is compact, then f can be extended to a continuous mapping f^∗ :D→D⁰.

Definition 9 ([2], Definition 2.3). We say that a ballB(a, r) in a geodesic metric spaceX is a GC-ballif any geodesic which connects any two pointsx and y inB[a, r] lies in B[a, r].

Definition 10 ([2], Definition 2.4).We say that a metric space X is a GC-geodesic space if the following conditions are satisfied:

(i) X is a geodesic metric space;

(ii) any ball B(a, r) with center a in X and radius r < diamX is a GC-ball.

Corollary 3. Suppose that X is a GC-geodesic metric space, Y is a Q-Loewner space, locally path connected (or geodesic), and letf :B(a, r)→D⁰ be a quasiconformal mapping, where B(a, r) is a ball with center a in X and radius r < diamX. If D⁰ is a QED domain and D⁰ is compact then f can be extended to a continuous mapping f^∗:D→D⁰.

Proof. Since X is a GC-geodesic metric space, by Proposition 2.4 [2], it follows that B(a, r) is locally arc connected on the boundary. Using Corol- lary 2, we get the desired conclusion.

To prove the next result, we recall the following proposition:

Remark 8 ([5], Corollary 4.11). Suppose that X and Y are Q-Ahlfors regular metric spaces with Q > 1, that X is a Loewner space and that Y is linearly locally connected. Then the inverse of a (metric) quasiconformal map f from X onto Y is (metric) quasiconformal.

Corollary 4. Suppose thatX,Y areQ-Ahlfors regular Loewner spaces (Q >1), locally path connected and f :D→D⁰ is a quasiconformal mapping, where D, D⁰ are QED. If D and D⁰ are compact sets, then f can be extended to a homeomorphism f^∗ :D→D⁰.

Proof. The proof follows by Theorems 1, 2, 3, Remarks 5, 7 and 8.

Remark 9 ([6], Theorem 3).In the last three results, we can consider proper and geodesic spaces which admit a (1, Q)-Poincar´e inequality instead ofQ-Loewner spaces. In order to prove this statement, we use the next result:

“Suppose that (X, d, µ) is a proper and geodesicQ-regular metric space, Q > 1. ThenX is a Loewner space if and only if it admits a (1, Q)-Poincar´e inequality.”

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Received 7 September 2013 “Spiru Haret” University, National College of Computer Science,

str. Zorilor 17, Suceava, Romania anca56@hotmail.com