In this section D denotes a bounded domain in ℝ n , n ≥ 3. Let
and Γ1 = ∂D \ Γ0. Using this notation we can state our main result.
Theorem 2.1. Assume is continuous on , harmonic and quasiconformal in D. Assume f is Hölder continuous with exponent α, 0 < α ≤ 1, on ∂D and Γ1is uniformly perfect. Then f is Hölder continuous with exponent α on .
If Γ0 is empty we obtain the following
Corollary 2.2. If is continuous on , Hölder continuous with exponent α, 0 < α ≤ 1, on ∂D, harmonic and quasiconformal in D and if ∂D is uniformly perfect, then f is Hölder continuous with exponent α on .
The first step in proving Theorem 2.1 is reduction to the case Γ0 = ø. In fact, we show that existence of a hqc extension of f across Γ0 follows from well known results. Let D' = D ∪ Γ0. Then D' is an open set in ℝ n , Γ0 is a closed subset of D' and ∂D' = Γ1.
Clearly cap(K ∩ Γ0) = 0 for each compact K ⊂ D', and therefore, by Lemma 7.14 in , Λ
(K ∩ Γ0) = 0 for each α > 0. In particular, Γ0 has σ-finite (n - 1)-dimensional Hausdorff measure. Since it is closed in D', we can apply Theorem 35.1 in  to conclude that f has a quasiconformal extension F across Γ0 which has the same quasiconformality constant as f.
Since Γ0 is a countable union of compact subsets K
of capacity zero and hence of Wiener capacity zero we conclude that Γ0 has Wiener capacity zero. Hence, by a classical result (see ), there is a (unique) extension of f which is harmonic in D'. Obviously, F = G is a harmonic quasiconformal extension of f to which has the same quasiconformality constant as f.
In effect, we reduced the proof of Theorem 2.1 to the proof of Corollary 2.2. We begin the proof of Corollary 2.2 with the following
Lemma 2.3. Let D ⊂ ℝ n be a bounded domain with uniformly perfect boundary. There exists a constant m > 0 such that for every y ∈ D we have
Proof. Fix y ∈ D as above and z ∈ ∂D such that |y - z| = d ≡ r. Clearly diam(∂D) = diam(D) > 2r. Set and, F3 = S(z, 2r). Let Γ
= Δ (F
; ℝ n ) for i, j = 1, 2, 3. By Järvi and Vorinen [4, Thm 4.1(3)], there exists a constant a = a(E, n) > 0 such that
while by standard estimates [2, 7.5] there exists b = b(n) > 0 such that
Next, by Vuorinen [8, Cor 5.41] there exists m = m(E, n) > 0 such that
Finally, with we have
In conclusion, from the above lemma, our assumption
and Lemma 8 in , we conclude that there is a constant M, depending on m, n, K(f), C and α only such that
However, an argument presented in  shows that the above estimate holds for y ∈ D, y ∈ ∂D without any further conditions, but with possibly different constant:
The following lemma was proved in  for real valued functions, but the proof relies on the maximum principle which holds also for vector valued harmonic functions, hence lemma holds for harmonic mappings as well.
Lemma 2.6. Assume is continuous on and harmonic in D. Assume for each x0 ∈ ∂D we have
Then |h(x) - h(y)| ≤ ω(|x - y|), whenever x, y ∈ D and |x - y| ≤ r0.
Now we combine (2.5) and the above lemma, with r0 = diam(D), to complete the proof of Corollary 2.2 and therefore of Theorem 2.1 as well.