Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 49–69

ON Q-HOMEOMORPHISMS

O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov

University of Helsinki, Department of Mathematics and Statistics P.O. Box 68, FI-00014 Helsinki, Finland; olli.martio@helsinki.fi NAS of Ukraine, Institute of Applied Mathematics and Mechanics 74 Roze Luxemburg St., 83114, Donetsk, Ukraine; ryaz@iamm.ac.donetsk.ua Technion – Israel Institute of Technology Haifa 32000, Israel; srebro@math.technion.ac.il Holon Academic Institute of Technology P.O. Box 305, Holon 58102, Israel; yakubov@hait.ac.il

Abstract. Space BMO-quasiconformal mappings satisfy a special modulus inequality that is used to define the class of Q-homeomorphisms. In this class we study distortion theorems, boundary behavior, removability and mapping problems. Our proofs are based on extremal length methods and properties of BMO functions.

1. Introduction

Let D be a domain in R^{n}, n ≥ 2, and let Q: D → [1, ∞] be a measurable function.

Definition 1.1. We say that a homeomorphism f: D → \overline{R^{n}} is a Q-homeo-morphism if

(1.2)  M(fΓ) ≤ ∫_{D} Q(x)ϱ^{n}(x) dm(x)

for every family Γ of paths in D and every admissible function ϱ for Γ.

Here we use only open paths γ: (a, b) → \overline{R^{n}}. We say that γ joins sets E and F in a domain D if γ(((a, b)) ⊂ D and γ is a restriction of a closed path \overline{γ}: [a, b] → R^{n} such that \overline{γ}(a) ∈ E and \overline{γ}(b) ∈ F. The family of all paths which join E and F in D will be denoted by Γ(E, F; D). Recall that, given a family of paths Γ in a domain D, a Borel function ϱ: R^{n} → [0, ∞] is called admissible for Γ, abbreviated ϱ ∈ adm Γ, if

(1.3)  ∫_{γ} ϱ(x) |dx| ≥ 1

2000 Mathematics Subject Classification: Primary 30C65; Secondary 30C75.

for each γ ∈ Γ. The (conformal) modulus M(Γ) of Γ is defined as

(1.4)  M(Γ) = inf_{ϱ∈adm Γ} ∫_{D} ϱ^{n}(x) dm(x).

An example of Q(x)-homeomorphisms is provided by a class of homeomorphisms f ∈ W_{loc}^{1,n}(D) whose dilatation majorant Q is in L_{loc}^{n−1}(D), see Theorem 2.19 below.

For f: D → R^{n} with partial derivatives a.e. and x ∈ D, we let f′(x) denote the Jacobian matrix of f at x or the differential operator of f at x, if it exists, by J(x) = J(x, f) = det f′(x) the Jacobian of f at x, and by |f′(x)| the operator norm of f′(x), i.e., |f′(x)| = max{|f′(x)h| : h ∈ R^{n}, |h| = 1}. We also let l(f′(x)) = min{|f′(x)h| : h ∈ R^{n}, |h| = 1}. The outer dilatation of f at x is defined by

(1.5)  K_{O}(x) = K_{O}(x, f) = \{ \begin{matrix}\frac{|f′(x)|^{n}}{|J(x, f)|}, & if J(x, f) ≠ 0, \\ 1, & if f′(x) = 0, \\ ∞, & otherwise, \end{matrix}

the inner dilatation of f at x by

(1.6)  K_{I}(x) = K_{I}(x, f) = \{ \begin{matrix}\frac{|J(x, f)|}{l(f′(x))^{n}} , & if J(x, f) ≠ 0, \\ 1, & if f′(x) = 0, \\ ∞, & otherwise, \end{matrix}

and the maximal dilatation, or in short the dilatation, of f at x by

(1.7)  K(x) = K(x, f) = max(K_{O}(x), K_{I}(x)),

cf. [MRV] and [Re_{1} ]. Note, that K_{I}(x) ≤ K_{O}(x)^{n−1} and K_{O}(x) ≤ K_{I}(x)^{n−1} , see e.g. 1.2.1 in [Re_{1} ], and, in particular, K_{O}(x), K_{I}(x) and K(x) are simultaneously finite or infinite. K(x, f) < ∞ a.e. is equivalent to the condition that a.e. either det f′(x) ≥ 0 or f′(x) = 0, cf. [GI] and [IŠ].

Definition 1.8. Given a function Q: D → [1, ∞], we say that a sensepreserving homeomorphism f: D → R^{n} is Q(x)-quasiconformal, abbr. Q(x)-qc, if f ∈W_{loc}^{1,n}(D) and

(1.9)  K(x, f) ≤ Q(x) a.e.

Definition 1.10. We say that f: D → R^{n} is BMO-quasiconformal, abbr. BMO-qc, if f is Q(x)-qc for some BMO function Q: D → [1, ∞].

On Q-homeomorphisms 51

Here BMO stands for the function space introduced by John and Nirenberg [JN], see also [RR]. Recall that a real-valued function φ ∈ L_{loc}^{1}(D) is said to be of bounded mean oscillation in D, abbr. φ ∈ BMO(D), if

(1.11)  ∥φ∥_{∗} = sup_{B⊂D} \frac{1}{|B|} ∫_{B} |φ(x) − φ_{B}| dx < ∞,

where the supremum is taken over all balls B in D and

(1.12)  φ_{B} = \frac{1}{|B|} ∫_{B} φ(x) dx

is the mean value of the function φ over B. It is well known thatL^{∞}(D) ⊂ BMO(D) ⊂ L_{loc}^{p}(D) for all 1 ≤ p < ∞.

Since L^{∞}(D) ⊂ BMO, the class of BMO-qc mappings obviously contains all qc mappings. We show that many properties of qc mappings hold for BMO-qc mappings. Note that Q-homeomorphisms, Q(x)-qc and BMO-qc mappings are Möbius invariants and hence the concepts extend to mappings f: D → \overline{R^{n}} = R^{n} ∪ {∞} as in the qc theory.

The subject of Q-homeomorphisms is interesting on its own right and has applications to much wider classes of mappings which we plan to investigate elsewhere. In this paper we study various properties as distortion, removability, boundary behavior and mapping properties of Q-homeomorphisms under various conditions on Q. Then the corresponding properties of Q(x)-qc mappings f: D → R^{n}, n ≥ 2, with Q ∈ L_{loc}^{n−1} are obtained as simple consequences of Theorem 2.19 below. A special attention is paid to BMO-qc mappings in R^{n}, n ≥ 3.

The study of related maps for n = 2 started by David [Da] and Tukia [Tu]. Recently Astala, Iwaniec, Koskela and Martin considered mappings with dilatation controlled by BMO functions for n ≥ 3, see e.g. [IKM] and [AIKM]. It is necessary to note the activity of the related investigations of mappings of finite distortion, see e.g. [KKM_{1} ], [KKM_{2} ], [IKO], [IKMS], [KR], [KO], [MV_{1} ] and [MV_{2} ]. The present paper is a continuation of our study of BMO-qc mappings in the plane [RSY_{1−3} ], cf. [IM], see also [Sa], and a similar geometric approach is used throughout.

For a, b ∈ \overline{R^{n}} and E, F ⊂ \overline{R^{n}} we let q(a, b), q(E) and q(E, F) denote the spherical (chordal) distance between the points a and b, the spherical diameter of E and the spherical distance between E and F, respectively. We denote by B^{n}(a, r) the Euclidean ball |x − a| < r in R^{n} with center a and radius r, S^{n}(a, r) = ∂B^{n}(a, r). We also let B^{n}(r) = B^{n}(0, r) and B^{n} = B^{n}(1), S^{n} = ∂B^{n}.

2. Preliminaries

2.1. Proposition. Let f: D → R^{n} be a Q(x)-qc mapping. Then

(i) f is differentiable a.e.,

(ii) f satisfies Lusin’s property (N),

(iii) J_{f}(x) ≥ 0 a.e. If, in addition, Q ∈ BMO(D), or if more generally Q ∈ L_{loc}^{n−1} , then f^{−1} ∈ W_{loc}^{1,n}(f(D)), and

(iv) f^{−1} is differentiable a.e.,

(v) f^{−1} has the property (N),

(vi) J_{f}(x) > 0 a.e.

Proof. (i) and (ii) follow from the corresponding results for W_{loc}^{1,n} homeomorphisms, see [Re_{2} ] and [Re_{3} ]. In view of (i) and the fact that f is sense-preserving, (iii) follows by Rado–Reichelderfer [RR^{∗}, p. 333].

Now, if Q ∈ BMO, then Q and hence K(x, f) belongs to L_{loc}^{p} for all p < ∞ and, in particular, to L_{loc}^{n−1} . Hence, by Theorem 6.1 in [HK], f^{−1} ∈ W_{loc}^{1,n}(f(D)) and thus (iv)–(vi) follow.

2.2. Lemma. Let Q be a positive BMO function in B^{n}, n ≥ 3, and let A(t) = {x ∈ R^{n} : t < |x| < e^{−1}}. Then for all t ∈ (0, e^{−2}),

(2.3)  ∫_{A(t)} \frac{Q(x) dm(x)}{(|x| log 1/|x|)^{n}} ≤ c

where c = c_{1}∥Q∥_{∗} + c_{2}Q_{1} , and c_{1} and c_{2} are positive constants which depend only on n. Here ∥Q∥_{∗} is the BMO norm of Q and Q_{1} is the average of Q over B^{n}(1/e).

Proof. Fix t ∈ (0, e^{−2}), and set
(2.4)  η(t) = ∫_{A(t)} \frac{Q(x) dm(x)}{(|x| log 1/|x|)^{n}} .

For k = 1, 2, …, write t_{k} = e^{−k}, A_{k} = {x ∈ R^{n} : t_{k+1} < |x| < t_{k}}, B_{k} = B^{n}(t_{k}) and let Q_{k} be the mean value of Q(x) in B_{k}. Choose an integer N, such that t_{N+1} ≤ t < t_{N}. Then A(t) ⊂ A(t_{N+1}) = ⋃_{k=1}^{N+1} A_{k}, and

(2.5)  η(t) ≤ ∫_{A(t_{N+1})} \frac{Q(x)}{|x|^{n} log^{n} 1/|x|} dx = S_{1} + S_{2}

where

(2.6)  S_{1} = ∑_{k=1}^{N+1} ∫_{A_{k}} \frac{Q(x) − Q_{k}}{|x|^{n} log^{n} 1/|x|} dx

On Q-homeomorphisms 53

and

(2.7)  S_{2} = ∑_{k=1}^{N+1} Q_{k} ∫_{A_{k}} \frac{dx}{|x|^{n} log^{n} 1/|x|}.

Since A_{k} ⊂ B_{k} and for x ∈ A_{k}, |x|^{−n} ≤ Ω_{n}e^{n}/|B_{k}|, where Ω_{n} = |B^{n}| and since log 1/|x| > k, it follows that

|S_{1}| ≤ Ω_{n} ∑_{k=1}^{N+1} \frac{1}{k^{n}} \frac{e^{n}}{|B_{k}|} ∫_{B_{k}} |Q(x) − Q_{k}| dx ≤ Ω_{n}e^{n}∥Q∥_{∗} ∑_{k=1}^{N+1} \frac{1}{k^{n}}

and, thus,

(2.8)  |S_{1}| ≤ 2Ω_{n}e^{n}∥Q∥_{∗}

because, for p ≥ 2,

(2.9)  ∑_{k=1}^{∞} \frac{1}{k^{p}} < 2.

To estimate S_{2} , we first obtain from the triangle inequality that

(2.10)  Q_{k} = |Q_{k}| ≤ ∑_{l=2}^{k} |Q_{l} − Q_{l−1}| + Q_{1}.

Next we show that, for l ≥ 2,

(2.11)  |Q_{l} − Q_{l−1}| ≤ e^{n}∥Q∥_{∗}.

Indeed,

\begin{aligned}|Q_{l} − Q_{l−1}| & = \frac{1}{|B_{l}|}|∫_{B_{l}}(Q(x) − Q_{l−1}) dx| \\ & ≤ \frac{e^{n}}{|B_{l−1}|} ∫_{B_{l−1}} |Q(x) − Q_{l−1}| dx ≤ e^{n}∥Q∥_{∗}. \end{aligned}

Thus, by (2.10) and (2.11),

(2.12)  Q_{k} ≤ Q_{1} + ∑_{l=2}^{k} e^{n}∥Q∥_{∗} ≤ Q_{1} + ke^{n}∥Q∥_{∗},

and, since

(2.13)  ∫_{A_{k}} \frac{dx}{|x|^{n} log^{n} 1/|x|} ≤ \frac{1}{k^{n}} ∫_{A_{k}} \frac{dx}{|x|^{n}} = ω_{n−1} \frac{1}{k^{n}} ,

where ω_{n−1} is the (n − 1)-measure of S^{n−1} , it follows that

S_{2} ≤ ω_{n−1} ∑_{k=1}^{N+1} \frac{Q_{k}}{k^{n}} ≤ ω_{n−1}Q_{1} ∑_{k=1}^{N+1} \frac{1}{k^{n}} + ω_{n−1}e^{n}∥Q∥_{∗} ∑_{k=1}^{N+1} \frac{1}{k^{(n−1)}} .

Thus, for n ≥ 3, we have by (2.9) that

(2.14)  S_{2} ≤ 2ω_{n−1}Q_{1} + 2ω_{n−1}e^{n}∥Q∥_{∗}.

Finally, from (2.8) and (2.14) we obtain (2.3), where c = c_{1}Q_{1} + c_{2}∥Q∥_{∗}, and c_{1} and c_{2} are constants which depend only on n.

2.15. Remark. For n ≥ 2, 0 < t < e^{−2} , and A(t) as in Lemma 2.2, let Γ_{t} denote the family of all paths joining the spheres |x| = t and |x| = e^{−1} in A(t). Then the function ϱ given by

(2.16)  ϱ(x) = \frac{1}{(log log 1/t)|x| log 1/|x|}

for x ∈ A(t) and ϱ(x) = 0 otherwise, belongs to adm Γ_{t}.

The following lemma provides the standard lower bound for the modulus of all paths joining two continua in R^{n}, see [Ge_{1} ], [Vu, 7.37]. The lemma involves the constant λ_{n} which depends only on n and which appears in the asymptotic estimates of the modulus of the Teichmüller ring R_{n}(t) = R^{n} \ ([−∞, 0] ∪ [t, ∞]).

2.17. Lemma. Let E and F be two continua in \overline{R^{n}}, n ≥ 2, with q(E) ≥ δ_{1} > 0 and q(F) ≥ δ_{2} > 0, and let Γ be the family of paths joining E and F. Then

(2.18)  M(Γ) ≥ \frac{ω_{n−1}}{(log \frac{2λ_{n}}{δ_{1}δ_{2}} )^{n−1}}

where ω_{n−1} is the (n − 1)-measure of S^{n−1} .

2.19. Theorem. Let f: D → R^{n} be a Q(x)-qc mapping with Q ∈ L_{loc}^{n−1}(D). Then, for every family Γ of paths in D and every ϱ ∈ adm Γ,

(2.20)  M(fΓ) ≤ ∫_{D} Q(x)ϱ^{n}(x) dm(x),

i.e., f is a Q-homeomorphism.

Proof. Since Q ∈ L_{loc}^{n−1} , we may apply Proposition 2.1. Thus f^{−1} ∈ W_{loc}^{1,n}(f(D)) and hence f^{−1} ∈ ACL_{loc}^{n}(f(D)), see e.g. [Maz, p. 8]. Therefore, by Fuglede’s theorem, see [Fu] and [Vä_{1} , p. 95], if Γ̃ is the family of all paths γ ∈ fΓ for which f^{−1} is absolutely continuous on all closed subpaths of γ, then M(fΓ) = M(Γ̃). Also, by Proposition 2.1, f^{−1} is differentiable a.e. Hence, as in the qc case, see [Vä, p. 110], given a function ϱ ∈ adm Γ, we let ϱ̃(y) = ϱ(f^{−1}(y))|(f^{−1})′(y)| for y ∈ f(D) and ϱ̃(y) = 0 otherwise. Then we obtain that for γ̃ ∈ Γ̃

∫γ̃ ϱ̃ds ≥ ∫_{f^{−1}∘γ̃} ϱ ds ≥ 1,

and consequently ϱ̃ ∈ adm Γ̃.

By Proposition 2.1, both f and f^{−1} are differentiable a.e. and have (N)- property and J(x, f) > 0 a.e., and since f^{−1} is a homeomorphism in W_{loc}^{1,n}(D), we can use the integral transformation formula and obtain

M(fΓ) = M(Γ̃) ≤ ∫_{f(D)} ϱ̃^{n} dm(y)

= ∫_{f(D)} ϱ(f^{−1}(y))^{n}|(f^{−1})′(y)|^{n} dm(y)

= ∫_{f(D)} \frac{ϱ(f^{−1}(y))^{n}}{l(f′(f^{−1}(y))^{n}} dm(y)

= ∫_{f(D)} ϱ(f^{−1}(y))^{n}K_{I}(f^{−1}(y), f)J(y, f^{−1}) dm(y)

≤ ∫_{f(D)} ϱ(f^{−1}(y))^{n}Q(f^{−1}(y))J(y, f^{−1}) dm(y)

= ∫_{D} Q(x)ϱ(x)^{n} dm(x).

The proof follows.

2.21. Corollary. Every BMO-qc mapping is a Q-homeomorphism with some Q ∈ BMO.

3. Distortion theorems

3.1. Theorem. Let f: B^{n} → \overline{R^{n}} be a Q-homeomorphism with Q ∈ BMO(B^{n}). If q(\overline{R^{n}} \ f(B^{n}(1/e))) ≥ δ > 0, then for all |x| < e^{−2}

(3.2)  q(f(x), f(0)) ≤ \frac{C}{(log 1/|x|)^{α}}

where C and α are positive constants which depend only on n, δ, the BMO norm ∥Q∥_{∗} of Q and the average Q_{1} of Q over the ball |x| < 1/e.

Proof. Fix t ∈ (0, e^{−2}). Let A(t), Γ_{t} and ϱ be as in Remark 2.15 and let δ_{t} = q(f(B^{n}(t))). Then, by Remark 2.15, ϱ ∈ adm Γ_{t}, and

(3.3)  M(fΓ_{t}) ≤ ∫_{R^{n}} Qϱ^{n} dm.

In view of (2.3),

(3.4)  ∫_{R^{n}} Qϱ^{n} dm = ∫_{A(t)} Qϱ^{n} dm ≤ \frac{c}{(log log 1/t)^{n−1}}

where c is the constant which appears in Lemma 2.2, see also Lemma 3.2 in [RSY_{2} ] for n = 2. On the other hand, Lemma 2.17 applied to M(fΓ_{t}) with E = f(\overline{B^{n}(t)}) and F = \overline{R^{n}} \ f(B^{n}(1/e)) yields

(3.5)  M(fΓ_{t}) ≥ \frac{ω_{n−1}}{(log \frac{2λ_{n}}{δδ_{t}} )^{n−1}} ,

and the result follows by (3.3)–(3.5) and the fact that q(f(x), f(0)) ≤ δ_{t} for |x| = t.

3.6. Corollary. Let ℱ be a family of Q-homeomorphisms f: D → R^{n}, with Q ∈ BMO(D), and let δ > 0. If every f ∈ ℱ omits two points a_{f} and b_{f} in R^{n} with q(a_{f}, b_{f}) ≥ δ, then ℱ is equicontinuous.

3.7. Theorem. Let f: B^{n} → \overline{R^{n}} be a Q-homeomorphism with Q ∈ L^{1}(B^{n}), f(0) = 0, q(\overline{R^{n}} \ f(B^{n})) ≥ δ > 0 and q(f(x_{0}), f(0)) ≥ δ for some x_{0} ∈ B^{n}. Then, for all |x| < r = min(|x_{0}|/2, 1 − |x_{0}|),

(3.8)  |f(x)| ≥ ψ(|x|)

where ψ(t) is a strictly increasing function with ψ(0) = 0 which depends only on the L^{1} -norm of Q in B^{n}, n and δ.

Proof. Given y_{0} with |y_{0}| < r choose a continuum E_{1} which contains the points 0 and x_{0} and a continuum E_{2} which contains the point y_{0} and ∂B^{n}, so that dist(E_{1}, E_{2} ∪ ∂B^{n}) = |y_{0}|. More precisely, denote by L the straight line generated by the pair of points 0 and x_{0} and by P the plane defined by the triple of the points 0, x_{0} and y_{0} (if y_{0} ∈ L, then P is an arbitrary plane passing through L). Let C be the circle under intersection of P and the sphere S^{n}(y_{0}, |y_{0}|) ⊂ B^{n}(|x_{0}|). Let t_{0} is the tangency point to C of the ray starting from x_{0} which is opposite to y_{0} with respect to L (an arbitrary one of the two possible if y_{0} ∈ L). Then E_{1} = [x_{0}, t_{0}] ∪ γ(0, t_{0}) where γ(0, t_{0}) is the shortest arc of C joining 0 and t_{0} , and E_{2} = [y_{0}, i_{0}] ∪ S^{n} where S^{n} = ∂B^{n} is the unit sphere and i_{0} is the point (opposite to t_{0} with respect to L) of the intersection of S^{n} with the straight line in P which is perpendicular to L and passing through y_{0} .

Let Γ denote the family of paths which join E_{1} and E_{2} . Then

ϱ(x) = |y_{0}|^{−1}χ_{B^{n}}(x) ∈ adm Γ

and hence,

(3.9)  M(fΓ) ≤ ∫ ϱ^{n}(x)Q(x) dm(x) ≤ |y_{0}|^{−n} ∫_{B^{n}} Q(x) dm(x) = \frac{∥Q∥_{1}}{|y_{0}|^{n}} .

The ring domain A′ = f(B^{n}\(E_{1} ∪ E_{2})) separates the continua E′_{1} = f(E_{1}) and E′_{2} = \overline{R^{n}}\f(B^{n}\E_{2}), and since

q(E′_{1}) ≥ q(f(x_{0}), f(0)) ≥ δ, q(E′_{2}) ≥ q(\overline{R^{n}}\f(B^{n})) ≥ δ

and

q(E′_{1}, E′_{2}) ≤ q(f(y_{0}), f(0))

it follows that

(3.10)  M(f(Γ)) ≥ λ(q(f(y_{0}), f(0)))

where λ(t) = λ_{n}(δ, t) is a strictly decreasing positive function with λ(t) → ∞ as t → 0, see [Vä_{1} , 12.7]. Hence, by (3.9) and (3.10),

|f(y_{0})| > q(f(y_{0}), f(0)) ≥ ψ(|y_{0}|)

where

ψ(t) = λ^{−1}(\frac{∥Q∥_{1}}{t^{n}} )

has the required properties.

3.11. Remark. In view of Theorem 2.19 and Corollary 2.21, Theorem 3.7 is valid for Q(x)-qc mappings with Q ∈ L^{n−1}(B^{n}), and Theorem 3.1 and Corollary 3.6 are valid for Q(x)-qc mappings with Q ∈ BMO(B^{n}).

4. Removability of isolated singularities

Here and in Theorems 4.1 and 4.5, we describe two different and unrelated cases where isolated singularities are removable.

4.1. Theorem. Let f: B^{n} \ {0} → R^{n} be a Q-homeomorphism. If

(4.2)  lim sup_{r→0} \frac{1}{|B^{n}(r)|} ∫_{B^{n}(r)} Q(x) dm(x) < ∞,

then f has an extension to B^{n} which is a Q-homeomorphism.

Proof. As the modulus of a family of paths which pass through the origin vanishes, it suffices to show that f has a continuous extension on B^{n}. Suppose that this is not the case. Since f is a homeomorphism, \overline{R^{n}} \ f(B^{n} \ \{0\}) consists of two connected compact sets F_{1} and F_{2} in \overline{R^{n}} where F_{1} contains the cluster set E = C(0, f) of f at 0. Here F_{1} is a nondegenerate continuum and using an arbitrary Möbius transformation we may assume that F_{1} ⊂ R^{n}. Now U = F_{1} ∪ f(B^{n} \ {0}) is a neighborhood of E. Thus there exists δ > 0 such that all balls B_{z} = B^{n}(z, δ), z ∈ F_{1} , are contained in U. Let V = ∪B_{z}.

Now, choose a point y ∈ F_{1} such that dist(y, ∂V) = δ, and a point z ∈ B_{y} \ F_{1} . Next, choose a path β: [0, 1] → B_{y} with β(0) = y, β(1) = z and β(t) ∈ B_{y} \F_{1} for t ∈ (0, 1]. Let α = f^{−1} ∘β. For r ∈ (0, |f^{−1}(z)|), let α_{r} denote the connected component of the curve α(I) \ B^{n}(r), I = [0, 1], which contains the point f^{−1}(z) = α(1), and let Γ_{r} denote the family of all paths joining α_{r} and the point 0 in B^{n} \ {0}. Then the function ϱ(x) = 1/r if x ∈ B^{n}(r) \ {0} and ϱ = 0 otherwise is in adm Γ_{r}, and by (4.2),

lim sup_{r→0} ∫_{B^{n}(r)\\{0\}} Q(x)ϱ^{n}(x) dm(x) = Ω_{n} lim sup_{r→0} \frac{1}{|B^{n}(r)|} ∫_{B^{n}(r)\\{0\}} Q(x) dm(x)

< ∞. (4.3) On the other hand, if we let Γ′_{r} denote the family of all paths joining two continua f(α_{r}) and E in B_{y} \ E, then Γ′_{r} ⊂ f(Γ_{r}), and thus

(4.4)  M(Γ′_{r}) ≤ M(fΓ_{r}).

Evidently, dist(f(α_{r}), E) → 0, and the diameter of f(α_{r}) increases as r → 0, and as both f(α_{r}) and E are subsets of a ball, M(fΓ_{r}) → ∞ as r → 0. This together with (4.3) and (4.4) contradicts the modulus inequality (2.20).

4.5. Theorem. Let f: B^{n} \ {0} → R^{n} be a Q-homeomorphism with Q ∈ BMO(B^{n} \ {0}). Then f has a Q(x)-homeomorphic extension to B^{n}.

Proof. Fix t ∈ (0, e^{−2}) and let A(t), Γ_{t} and ϱ be as in Remark 2.15. Then, by Lemma 2.17,

(4.6)  \frac{ω_{n−1}}{(log \frac{2λ_{n}}{δδ_{t}} )^{n−1}} ≤ M(fΓ_{t}) ≤ ∫_{A(t)} Qϱ^{n} dm,

where δ = q(f(∂B^{n}(e^{−1}))) and δ_{t} = q(f(∂B^{n}(t))). Since isolated singularities are removable for BMO functions, see [RR], we may assume that Q is defined in B^{n} and that Q ∈ BMO(B^{n}). Thus, by Lemma 3.2 in [RSY_{2} ] for n = 2 and Lemma 2.2 for n ≥ 3

(4.7)  ∫_{A(t)} Q(x)ϱ^{n} dm ≤ \frac{c}{(log log 1/t)^{n−1}} .

Since here c depends only on n, ∥Q∥_{∗} and Q_{1} = Q_{B^{n}(1/e)} , we obtain by (4.6)– (4.7) that δ_{t} → 0 as t → 0, and hence that lim_{x→0} f(x) exists.

4.8. Corollary. If f: R^{n} → R^{n} is a BMO-qc mapping, then f has a homeomorphic extension to \overline{R^{n}} and, in particular, f(R^{n}) = R^{n}.

5. Boundary behavior

For the boundary behavior some regularity of the boundary is needed for which the following notation is used.

We say that a domain D in R^{n} is a BMO extension domain if every u ∈ BMO(D) has an extension to R^{n} which belongs to BMO(R^{n}). It was shown in [GO] and [Jo] that a domain D is a BMO extension domain if and only if D is a uniform domain, i.e., for some a and b > 0, each pair of points x_{1}, x_{2} ∈ D can be joined by a rectifiable arc γ ⊂ D such that

(5.1)  l(γ) ≤ a · |x_{1} − x_{2}|

and, for all x ∈ γ,

(5.2)  min_{i=1,2} l(γ(x_{i}, x)) ≤ b · dist(x, ∂D)

where l(γ) is the Euclidean length of γ, γ(x_{i}, x) is the part of γ between x_{i} and x.

The uniform domains were introduced in [MS] and their various characterizations can be found in [Ge_{2} ], [Ma], [Mar], [Vä_{2} ] and [Vu]. It was shown in [MS], p. 387, that uniform domains are invariant under quasiconformal mappings of R^{n}. In particular, every domain which is bounded by a quasisphere, i.e., the image of ∂B^{n} under a qc automorphism of R^{n}, is uniform. Note that a bounded convex domain is uniform. We also write u ∈ BMO( \overline{D}) if u has a BMO extension to an open set U ⊂ R^{n} such that \overline{D} ⊂ U. Domains D in R^{n} for which every u ∈ BMO(D) admits such extension can be characterized as relatively uniform domains, see e.g. [Go_{2} ].

A domain D ⊂ R^{n} is called a quasiextremal distance domain or a QED domain if there is K ≥ 1 such that, for each pair of disjoint continua E and F in D,

(5.3)  M(Γ(E, F; \overline{R^{n}})) ≤ K · M(Γ(E, F; D)).

It is known that every uniform domain D is a QED domain and there exist QED domains which are not uniform, see [GM], pp. 189 and 194. Every QED domain D is quasiconvex, i.e., (5.1) holds for all x_{1} and x_{2} ∈ D \ {∞}, see Lemma 2.7 in [GM], p. 184. Hence every QED domain D is locally connected at ∂D, i.e., every point x ∈ ∂D has an arbitrarily small neighborhood U such that U ∩ D is connected, cf. also Lemma 2.11 in [GM], p. 187, and [HK_{1} ], p. 190. Note that every Jordan domain D in R^{n} is locally connected at ∂D, see [Wi], p. 66.

We say that ∂D is strongly accessible if, for nondegenerate continua E and F in \overline{D},

(5.4)  M(Γ(E, F; D)) > 0,

and that ∂D is weakly flat if, for nondegenerate continua E and F in \overline{D} with E ∩ F ≠ ∅,

(5.5)  M(Γ(E, F; D)) = ∞.

These properties are clearly invariant under qc mappings of \overline{R^{n}} and they are closely related to properties P_{1} and P_{2} by Väisälä in [Vä_{1} , 17.5] as well as to the notions of quasiconformal flatness and quasiconformal accessibility by Näkki in [Nä] and by Herron and Koskela in [HK_{1} ].

We shall show below that weak flatness implies strong accessibility. The converse does not hold as in the example of a disk minus a cut. Note that the conditions (5.4) and (5.5) automatically hold if E and F are inside D, see e.g. Lemma 1.15 in [Nä], p. 16, and [Vä_{1} , 10.12], but not if E and F are in ∂D. It is known that for a QED domain the inequality (5.3) holds for each pair of disjoint continua E and F in \overline{D}, see Theorem 2.8 in [HK_{2} ], p. 173, cf. Lemma 6.11 in [MV], p. 35. The latter property of QED domains also implies (5.3) for nondegenerate intersecting continua E and F in \overline{D}. Hence QED domains and, consequently, uniform domains have weakly flat boundaries, cf. Lemma 3.1 in [HK_{1} ], p. 196.

5.6. Lemma. If the boundary of a domain D in \overline{R^{n}}, n ≥ 2, is weakly flat, then it is strongly accessible.

Proof. Let E and F be nondegenerate continua in \overline{D}. Without loss of generality we may assume that E ∩ F = ∅, that ∞ lies outside of E ∪ F and that E ⊂ ∂D, see Lemma 1.15 in [Nä], p. 16. Take 0 < ε < \frac{1}{2} dist(E, F). Then E_{ε} ∩ F_{ε} = ∅ where E_{ε} = {x ∈ D : dist(x, E) < ε} and F_{ε} = {x ∈ D : dist(x, F) < ε}. Since each path in Γ(E, E; D) contains a subpath which belongs to Γ(E, E_{ε}; D), it follows, see e.g. [Fu], p. 178, cf. [AB], p. 115, that M(Γ(E, E; D)) ≤ M(Γ(E, E_{ε}; D)). In view of the weak flatness of ∂D, M(Γ(E, E; D)) = ∞. Therefore M(Γ(E, E_{ε}; D)) = ∞. By the Lindelöf principle, see e.g. [Ku], p. 54, the open subsets E_{ε} and F_{ε} of D can be covered by countable collections of open and, consequently, closed balls B inside E_{ε} and F_{ε}, respectively. Thus, by countable subadditivity of the modulus we can find a couple of closed balls, B_{0} ⊂ E_{ε} and B_{0}^{∗} ⊂ F_{ε}, B_{0} ∩ B_{0}^{∗} = ∅, such that

(5.7)  M(Γ(E, B_{0}; D)) > 0

and

(5.8)  M(Γ(F, B_{0}^{∗}; D)) > 0,

see Theorem 1 in [Fu], p. 176. Also

(5.9)  M(Γ(B_{0}, B_{0}^{∗}; D)) > 0,

see e.g. Lemma 1.15 in [Nä].

Now, we use the idea of Näkki which appears in the proof of Theorem 1.16 in [Nä]. Let ϱ ∈ adm Γ(E, B_{0}^{∗}; D). If

(5.10)  ∫_{γ} ϱ ds ≥ \frac{1}{3} or ∫_{γ′} ϱ ds ≥ \frac{1}{3}

for all rectifiable paths γ ∈ Γ(E, B_{0}; D) and γ′ ∈ Γ(B_{0}, B_{0}^{∗}; D), respectively, then 3ϱ ∈ adm Γ(E, B_{0}; D) or 3ϱ ∈ adm Γ(B_{0}, B_{0}^{∗}; D), and hence by (5.7) and (5.9)

(5.11)  ∫_{D} ϱ^{n} dm ≥ 3^{−n} min(M(Γ(E, B_{0}; D)), M(Γ(B_{0}, B_{0}^{∗}; D))) > 0.

If (5.11) does not hold for a couple of such paths γ and γ′, then

(5.12)  ∫_{α} ϱ ds > \frac{1}{3}

for every rectifiable path α ∈ Γ(γ, γ′; R_{0}) where R_{0} is a ring r_{0} < |x−c_{0}| < r′_{0} , c_{0} and r_{0} are the center and the radius of B_{0} , respectively, and r′_{0} = r_{0}+dist(B_{0}, B_{0}^{∗}∪ ∂D), i.e., 3ϱ ∈ adm Γ(γ, γ′; R_{0}), and hence

(5.13)  ∫_{D} ϱ^{n} dm ≥ 3^{−n}c_{n} log \frac{r′_{0}}{r_{0}} > 0,

see [Vä_{1} , 10.12]. Thus, by (5.11) and (5.13)

(5.14)  M(Γ(E, B_{0}^{∗}; D)) > 0.

If I = IntB_{0}^{∗} ∩ F ≠ ∅, then by Lemma 1.15 in [Nä] M(Γ(B_{0}, Ī; D)) > 0, and arguing as above (take Ī = B_{0}^{∗} ∩ F instead of B_{0}^{∗}) we obtain M(Γ(E, Ī; D)) > 0 and hence by monotonicity of the modulus

(5.15)  M(Γ(E, F; D)) > 0.

If B_{0}^{∗} ∩ F = ∂B_{0}^{∗} ∩ F ≠ ∅, then by subadditivity of the modulus we obtain (5.8), (5.9) and (5.14) for another ball B′_{0} ⊂ IntB_{0}^{∗} with B′_{0} ∩ F = ∅. Finally, if B_{0}^{∗} ∩ F = ∅, then repeating the above arguments (as in the proof of (5.14), replace B_{0} and B_{0}^{∗} by B_{0}^{∗} and F, respectively) we again obtain (5.15) by (5.8) and (5.14).

5.16. Lemma. Let D be a domain in R^{n}, n ≥ 2, which is locally connected at ∂D and let f: D → D′ ⊂ R^{n} be a Q-homeomorphism onto D′ with Q ∈ BMO( \overline{D}). If ∂D′ is strongly accessible, then f has a continuous extension f̃: \overline{D} → \overline{D}′.

Proof. Let x_{0} ∈ ∂D. As BMO functions and Q-homeomorphisms are Möbius invariant, we may assume that x_{0} = 0 and that ∂B^{n} ∩ D ≠ ∅. We will show that the cluster set E = C(0, f) of f at 0 is a point, which will prove that f(x) has a limit at x_{0} .

Since D is locally connected at 0, E is a continuum. Suppose that E is nondegenerate. For t ∈ (0, 1/e), let D_{t} denote the component of B^{n}(t) ∩ D for which 0 ∈ \overline{D}_{t}. Note, that D_{t} is well defined, since D is locally connected at 0, and that D_{t} ⊂ D_{t′} for t < t′. For t ∈ (0, e^{−2}], let Γ_{t} denote the family of path joining \overline{D_{t}} and the set S = D ∩ ∂B^{n}(1/e) in D_{1/e} \ \overline{D}_{t}. As in Lemma 2.2, we let A(t) denote the spherical ring t < |x| < 1/e. Then the function ϱ(x) defined in Remark 2.15 is admissible for Γ_{t}, and hence

(5.17)  M(fΓ_{t}) ≤ ∫_{D} Q(x)ϱ^{n}(x) dm(x).

If Q ∈ BMO( \overline{D}), we may apply (3.3) in [RSY_{2} ] for n = 2 and (2.3) for n ≥ 3, and get

(5.18)  ∫_{D} Q(x)ϱ^{n}(x) dm(x) ≤ ∫_{A(t)} Q(x)ϱ^{n}(x) dm(x) → 0

as t → 0. On the other hand

(5.19)  M(fΓ_{t}) ≥ M(Γ(\overline{f(S)}, E; D′)).

Now, ∂D′ is strongly accessible, f(S) contains a nondegenerate continuum and E is nondegenerate. Therefore, the right-hand side in (5.19) is positive contradicting (5.17) and (5.18). This shows that the cluster set of f at every point of ∂D is degenerate and thus f has a continuous extension on \overline{D}.

5.20. Lemma. Let D be a domain in R^{n} and f a Q-homeomorphism of D onto a domain D′ in R^{n} with Q ∈ L^{1}(D). Suppose that D is locally connected at ∂D. If ∂D′ is weakly flat, then C(x_{1}, f) ∩ C(x_{2}, f) = ∅ for every two distinct points x_{1} and x_{2} in ∂D.

Proof. With no loss of generality we may assume that the domain D is bounded. For i = 1, 2, let E_{i} denote the cluster sets C(x_{i}, f) and suppose that E_{1} ∩ E_{2} ≠ ∅. Write d = |x_{1} − x_{2}|. Since D is locally connected in ∂D, there are neighborhoods U_{i} of x_{i}, such that W_{i} = D∩U_{i} is connected and U_{i} ⊂ B^{n}(x_{i}, d/3), i = 1, 2. Then the function ϱ(x) = 3/d if x ∈ D∩B^{n}((x_{1}+x_{2})/2, d) and ϱ(x) = 0 elsewhere is admissible for the family Γ = Γ(\overline{W}_{1}, \overline{W}_{2}; D). Thus,

(5.21)  M(fΓ) ≤ ∫_{D} Q(x)ϱ^{n}(x) dm(x) ≤ \frac{3^{n}}{d^{n}} ∫_{D} Q(x) dm(x) < ∞.

On the other hand

(5.22)  M = M(Γ(E_{1}, E_{2}; D)) ≤ M(fΓ).

But as ∂D′ is weakly flat, and E_{i}, i = 1, 2, are nondegenerate continua in \overline{D′} with non-empty intersection, M = ∞, contradicting (5.21). The assertion follows.

5.23. Corollary. Let D, D′, f and Q be as in Lemma 5.20. Then f^{−1} has a continuous extension to \overline{D′}.

5.24. Corollary. Let E be a nondegenerate continuum in B^{n} and Q ∈ L^{1}(B^{n} \ E). Then there exists no Q-homeomorphism of B^{n} \ E onto B^{n} \ {0}. If Q ∈ L^{n−1}(B^{n} \ E), then there exists no Q(x)-qc mapping of B^{n} \ E onto B^{n} \ {0}.

5.25. Corollary. Let f: D → D′ ⊂ R^{n} be a Q-homeomorphism onto D′ with Q ∈ BMO( \overline{D}). If D locally connected at ∂D and ∂D′ is weakly flat, then f has a homeomorphic extension f̃: \overline{D} → \overline{D′}.

5.26. Theorem. Let f: D → D′ be a Q-homeomorphism between QED domains D and D′ with Q ∈ BMO( \overline{D}). Then f has a homeomorphic extension f̃: \overline{D} → \overline{D′}.

This and the next theorem extend the known Gehring–Martio results, see [GM], p. 196, and [MV], p. 36, from qc mappings to Q-homeomorphisms with Q ∈ BMO( \overline{D}) and to BMO-qc mappings, respectively.

5.27. Theorem. Let f: D → D′ be a BMO-qc mapping between uniform domains D and D′. Then f has a homeomorphic extension f̃: \overline{D} → \overline{D′}.

5.28. Corollary. Let f: D → D′ be a BMO-qc mapping between bounded convex domains D and D′. Then f has a homeomorphic extension f̃: \overline{D} → \overline{D′}.

5.29. Corollary. If D is a domain in R^{n} which is locally connected at ∂D and if D is not a Jordan domain, then D cannot be mapped onto B^{n} by a Q-homeomorphism with Q ∈ BMO( \overline{D}).

5.30. Corollary. If a domain D in R^{n} is uniform but not Jordan, then there is no BMO-qc mapping of D onto B^{n}.

In 7.2 below we show that for every n ≥ 3 there is a bounded uniform domain in R^{n} which is a topological ball and not Jordan.

6. Mapping problems

In Section 4, we showed that there are no BMO-qc mappings of R^{n} onto a proper subset of R^{n}, nor BMO-qc mappings of a punctured ball onto a domain that has two nondegenerate boundary components. We may consider the following two questions.

(a) Are there any proper subsets of R^{n} that can be mapped BMO-quasicon-formally onto R^{n}?

(b) Are there any nondegenerate continua E in B^{n} such that B^{n} \ E can be mapped BMO-quasiconformally onto B^{n} \ {0}?

In [RSY_{2} ] we showed that the answer to these questions is negative if n = 2. The proofs were based on the Riemann Mapping Theorem and on the existence of a homeomorphic solution to the Beltrami equation

wz̄ = μ(z)w_{z}

for measurable functions μ with ∥μ∥_{∞} ≤ 1 which satisfy

\frac{(1 + |μ(z)|)}{(1 − |μ(z)|)} ≤ Q(z)

a.e. for some BMO function Q. One may modify Questions (a) and (b) by replacing the words “BMO-quasiconformally” by ”by a Q-homeomorphism”. Below, we provide a negative answer to Questions (a) and (b) in some special cases when n > 2.

We say that a proper subdomain D of R^{n} is an L^{1} -BMO domain if u ∈ L^{1}(D) whenever u ∈ BMO(D). Evidently, D is an L^{1} -BMO domain, if D is a bounded uniform domain. By [Sta], pp. 106–107, cf. [Go_{1} ], p. 69, D is an L^{1} - BMO domain if and only if k_{D}( · , x_{0}) ∈ L^{1}(D) where k_{D} is the quasihyperbolic metric on D,

(6.1)  k_{D}(x, x_{0}) = inf_{γ} ∫_{γ} \frac{ds}{d(y, ∂D)}

where ds denotes the Euclidean length element, d(y, ∂D) the Euclidean distance from y ∈ D to ∂D, and the infimum is taken over all rectifiable paths γ joining x to x_{0} in D. L^{1} -BMO domains are not invariant under quasiconformal mappings of R^{n}, however, they are invariant under quasi-isometries, see [Sta], pp. 119 and 112.

In particular, every John domain is an L^{1} -BMO domain, see Theorem 3.14 in [Sta], p. 115. A domain D ⊂ R^{n} is called a John domain if there exist 0 < α ≤ β < ∞ and a point x_{0} ∈ D such that, for every x ∈ D, there is a rectifiable path γ: [0, l] → D parametrized by arclength such that γ(0) = x, γ(l) = x_{0} , l ≤ β and

(6.2)  d(γ(t), ∂D) ≥ \frac{α}{l} · t

for all t ∈ [0, l]. A John domain need not be uniform but a bounded uniform domain is a John domain, see [MS], p. 387. Note also that John domains are invariant under qc mappings of R^{n}, see [MS], p. 388. A convex domain D is a John domain if and only if D is bounded. For characterizations of John domains, see [He], [MS] and [NV].

Hölder domains are also L^{1} -BMO domains. A domain D ⊂ R^{n} is said to be a Hölder domain if there exist x_{0} ∈ D, δ ≥ 1 and C > 0 such that

(6.3)  k_{D}(x, x_{0}) ≤ C + δ · log \frac{d(x_{0}, ∂D)}{d(x, ∂D)}

for all x ∈ D. It is known that D is a Hölder domain if and only if the quasihyperbolic metric k_{D}(x, x_{0}) is exponentially integrable in D, see [SS]. Thus, a Hölder domain is also an L^{1} -BMO domain.

6.4. Theorem. Let D be a domain in R^{n}, D ≠ R^{n}, n ≥ 2, and f: D → R^{n} a Q-homeomorphism. If there exist a point b ∈ ∂D and a neighborhood U of b such that Q|_{D∩U} ∈ L^{1} , then f(D) ≠ R^{n}.

Proof. The statement is trivial if D is not a topological ball. Suppose that D is a topological ball. By the Möbius invariance, we may assume that b = 0 and that ∞ ∈ ∂D. Let r > 0 be such that B^{n}(r) ⊂ U. Then Q is integrable in B^{n}(r) ∩ D. Choose two arcs E and F in B^{n}(r/2) ∩ D each having exactly one end point in ∂D such that 0 < dist(E, F) < r/2. Such arcs exist. Indeed, since ∂D is connected and 0 and ∞ belong to ∂D, the sphere ∂B^{n}(r/2) meets ∂D and contains a point x_{0} which belongs to D. Then one can take E as a maximal line segment in (0, x_{0}] ∩ D with one end point at x_{0} and the other one in ∂D, and F as a circular arc in the maximal spherical cap in ∂B^{n}(r/2) ∩ D which is centered at x_{0} , so that F has one end point in ∂D and the other one in D.

Now, let Γ denote the family of all paths which join E and F in D. Then ϱ(x) = dist(E, F)^{−1} if x ∈ B^{n}(r) ∩ D and ϱ(x) = 0 otherwise is admissible for Γ. Then by (1.2)

(6.5)  M(fΓ) ≤ ∫_{D} Qϱ^{n} dm ≤ \frac{1}{dist(E, F)^{n}} ∫_{B^{n}(r)∩D} Q dm < ∞.

On the other hand, if f(D) = R^{n}, then f(E) and f(F) meet at ∞ and fΓ is the family of paths joining f(E) and f(F) in R^{n}. Thus M(fΓ) = ∞. Contradiction showing that f(D) ≠ R^{n}.

As a consequence of Theorem 6.4, we have the following corollaries which say that a proper subdomain D of R^{n} having a nice boundary at least at one point of ∂D cannot be mapped BMO-quasiconformally onto R^{n}.

6.6. Corollary. Let D be a domain in R^{n}, D ≠ R^{n}, n ≥ 2, and let f: D → R^{n} be a Q-homeomorphism with Q ∈ BMO(D). If there exists a point b ∈ ∂D and a neighborhood U of b such that D ∩ U is an L^{1}- BMO domain or, in particular, if ∂(D ∩ U) is a quasisphere, then f(D) ≠ R^{n}.

6.7. Remark. Theorem 6.4 implies in particular that, if a BMO-qc mapping f of D is onto R^{n}, then either D = R^{n} or the domain D cannot be (even locally at any boundary point) convex, uniform, John or Hölder.

By the techniques which are used in the proof of Theorem 6.4, one can establish the following theorem which gives partial answers to (b).

6.8. Theorem. Let E be a nondegenerate continuum in B^{n}, D = B^{n} \ E, and f: D → R^{n} a Q-homeomorphism. If there exist a point x_{0} ∈ ∂D ∩ B^{n} and a neighborhood U of x_{0} such that Q|_{D∩U} ∈ L^{1} , then f(D) is not a punctured topological ball.

6.9. Corollary. Let E be a nondegenerate continuum in B^{n} and D = B^{n} \ E. If there exist a point x_{0} ∈ ∂D ∩ B^{n} and a neighborhood U of x_{0} such that U \ E is an L^{1}- BMO domain or, in particular, if ∂(U \ E) is a quasisphere, then D cannot be mapped BMO-quasiconformally onto B^{n} \ {0}.

6.10. Remark. The condition Q|_{D∩U} ∈ L^{1} which appears in Theorems 6.4 and 6.8 holds for Q ∈ BMO(D) if k_{D∩U} ∈ L^{1} and |∂D ∩ U| > 0, see [Sta]. Note that the latter property is impossible for convex, uniform, QED as well as for John domains, see [Ma], p. 204, [GM], p. 189, and [MV], p. 33.

7. Some examples

We say that a domain D in R^{n}, n ≥ 2, is a quasiball, respectively, BMOquasiball if there exists a homeomorphism of D onto B^{n} which is qc, respectively, BMO-qc. We say that a set S in \overline{R^{n}} is a quasisphere, respectively, BMOquasisphere if there exists a qc mapping, respectively, BMO-qc mapping f of \overline{R^{n}} onto itself such that f(S) = ∂B^{n}.

The following example shows that there is a BMO-quasicircle γ which is not a quasicircle.

7.1. Example. Consider the curve γ = γ_{1} ∪ γ_{2} ∪ γ_{3} where γ_{1} = [0, ∞], γ_{2} = [−∞, −1/e] and

γ_{3} = {te^{iπ/ log 1/t} : 0 < t < 1/e}.

Clearly, γ does not satisfy Ahlfors’s three points condition, and hence it is not a quasicircle. However, γ is a BMO-quasicircle. Indeed, the map f: C → C which is identity in C \ B^{2} and is given for |z| < 1 by

f(re^{iθ}) = \{ \begin{matrix}r exp i(θ log 1/r), & if 0 ≤ θ ≤ \frac{π}{log 1/r}, \\ r exp iπ(1 + \frac{1 − θ/π log 1/r}{1 − 2 log 1/r} ), & if \frac{π}{log 1/r} ≤ θ < 2π \end{matrix}

is Q(z)-qc with Q(re^{iθ}) = max(1, log 1/r) which is BMO-qc in C and maps γ onto R.

Note that R^{n} is a topological ball which cannot be mapped by a BMO-qc mapping onto B^{n}, see Corollary 4.8. In view of Corollary 5.30, the following example shows that, for every n ≥ 3, there exists a proper subdomain of B^{n} which is a topological ball but not a BMO-quasiball.

7.2. Example. Let B = B^{n} \ C^{n}(ε) where C^{n}(ε) is a cone in B^{n} with vertex v = ∂B^{n} ∩{x_{n} = 1} and base B^{n}(ε)∩{x_{n} = 0}, 0 < ε < 1. For n ≥ 3, the domain B is uniform. Evidently B is a topological ball. However, the boundary of B is not homeomorphic to the sphere S^{n−1} because the point v splits ∂B into two components.

Acknowledgements. The first two authors were supported by a grant from the Academy of Finland. The research of the second author was partially supported by grants from the University of Helsinki and from Technion – Israel Institute of Technology as well as by Grant 01.07/00241 of SFFI of Ukraine; the research of the third author was partially supported by a grant from the Israel Science Foundation and by Technion Fund for the Promotion of Research, and the fourth author was partially supported by a grant from the Israel Science Foundation.

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Received 29 August 2002