In this section *D* denotes a bounded domain in ℝ ^{n} , *n* ≥ 3. Let

{\Gamma}_{0}=\left\{x\in \partial D:\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{cap}}\phantom{\rule{2.77695pt}{0ex}}\left(\overline{B}\left(x,\epsilon \right)\cap \partial D\right)=0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{some}}\phantom{\rule{2.77695pt}{0ex}}\epsilon >0\right\},

and Γ_{1} = *∂D* \ Γ_{0}. Using this notation we can state our main result.

**Theorem 2.1**. *Assume* f:\overline{D}\to {\mathbb{R}}^{n} is continuous on \overline{D}, *harmonic and quasiconformal in D. Assume f is Hölder continuous with exponent α*, 0 < *α* ≤ 1, *on ∂D and* Γ_{1}*is uniformly perfect. Then f is Hölder continuous with exponent α on* \overline{D}.

If Γ_{0} is empty we obtain the following

**Corollary 2.2**. *If* f:\overline{D}\to {\mathbb{R}}^{n} is continuous on \overline{D}, *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* \overline{D}.

The first step in proving Theorem 2.1 is reduction to the case Γ_{0} = ø. In fact, we show that existence of a hqc extension of f across Γ_{0} follows from well known results. Let *D*' = *D* ∪ Γ_{0}. Then *D*' is an open set in ℝ ^{n} , Γ_{0} is a closed subset of *D*' and *∂D*' = Γ_{1}.

Clearly cap(*K* ∩ Γ_{0}) = 0 for each compact *K* ⊂ *D*', and therefore, by Lemma 7.14 in [8], Λ_{
α
}(*K* ∩ Γ_{0}) = 0 for each *α* > 0. In particular, Γ_{0} has *σ*-finite (*n* - 1)-dimensional Hausdorff measure. Since it is closed in *D*', we can apply Theorem 35.1 in [2] to conclude that *f* has a quasiconformal extension *F* across Γ_{0} which has the same quasiconformality constant as *f*.

Since Γ_{0} is a countable union of compact subsets *K*_{
j
} of capacity zero and hence of Wiener capacity zero we conclude that Γ_{0} has Wiener capacity zero. Hence, by a classical result (see [11]), there is a (unique) extension G:\overline{{D}^{\prime}}\to {\mathbb{R}}^{n} of *f* which is harmonic in *D*'. Obviously, *F* = *G* is a harmonic quasiconformal extension of *f* to \overline{{D}^{\prime}} 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*

\mathsf{\text{cap}}\left(\overline{B}\left(y,\frac{d}{2}\right),D\right)\phantom{\rule{2.77695pt}{0ex}}\ge m,\phantom{\rule{1em}{0ex}}d=\mathsf{\text{dist}}\left(y,\phantom{\rule{2.77695pt}{0ex}}\partial D\right).

(2.4)

*Proof*. Fix *y* ∈ *D* as above and *z* ∈ *∂D* such that |*y* - *z*| = *d* ≡ *r*. Clearly *diam*(*∂D*) = *diam*(*D*) > 2*r*. Set {F}_{1}=\overline{B}\left(z,r\right)\cap \left(\partial D\right) and{F}_{2}=\overline{B}\left(z,r\right)\cap \overline{B}\left(y,\frac{d}{2}\right), *F*_{3} = *S*(*z*, 2*r*). Let Γ _{
i,j
} = Δ (*F*_{
i
} , *F*_{
j
} ; ℝ ^{n} ) for *i, j* = 1, 2, 3. By Järvi and V\stackrel{\xb0}{\mathsf{\text{u}}}orinen [4, Thm 4.1(3)], there exists a constant *a* = *a*(*E*, *n*) > 0 such that

M\left({\Gamma}_{1,3}\right)\ge a

while by standard estimates [2, 7.5] there exists *b* = *b*(*n*) > 0 such that

M\left({\Gamma}_{2,3}\right)\ge b.

Next, by Vuorinen [8, Cor 5.41] there exists *m* = *m*(*E*, *n*) > 0 such that

M\left({\Gamma}_{1,2}\right)\ge m.

Finally, with B=\overline{B}\left(y,d2\right) we have

\mathsf{\text{cap}}\left(B,D\right)=M\left(\Delta \left(B,\partial D;{\mathbb{R}}^{n}\right)\right)\ge M\left({\Gamma}_{1,2}\right)\ge m.

In conclusion, from the above lemma, our assumption

|f\left({x}_{1}\right)-f\left({x}_{2}\right)|\phantom{\rule{2.77695pt}{0ex}}\le C|{x}_{1}-{x}_{2}{|}^{\alpha},\phantom{\rule{1em}{0ex}}{x}_{1},{x}_{2}\in \partial D,

and Lemma 8 in [1], we conclude that there is a constant *M*, depending on *m*, *n*, *K*(*f*), *C* and *α* only such that

|f\left(x\right)-f\left(y\right)|\phantom{\rule{2.77695pt}{0ex}}\le M|x-y{|}^{\alpha},\phantom{\rule{1em}{0ex}}y\in D,\phantom{\rule{2.77695pt}{0ex}}x\in \partial D,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{dist}}\left(y,\partial D\right)=\phantom{\rule{2.77695pt}{0ex}}|x-y|.

However, an argument presented in [1] shows that the above estimate holds for *y* ∈ *D*, *y* ∈ *∂D* without any further conditions, but with possibly different constant:

|f\left(x\right)-f\left(y\right)|\phantom{\rule{2.77695pt}{0ex}}\le {M}^{\prime}|x-y{|}^{\alpha},\phantom{\rule{1em}{0ex}}y\in D,\phantom{\rule{2.77695pt}{0ex}}x\in \partial D.

(2.5)

The following lemma was proved in [12] 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* h:\overline{D}\to {\mathbb{R}}^{n} is continuous on \overline{D} *and harmonic in D. Assume for each x*_{0} ∈ *∂D we have*

\underset{{B}_{r}\left({x}_{0}\right)\cap {D}^{\prime}}{sup}|h\left(x\right)-h\left({x}_{0}\right)|\phantom{\rule{2.77695pt}{0ex}}\le \omega \left(r\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}0<r\le {r}_{0}.

Then |*h*(*x*) - *h*(*y*)| ≤ *ω*(|*x* - *y*|), whenever *x*, *y* ∈ *D* and |*x* - *y*| ≤ *r*_{0}.

Now we combine (2.5) and the above lemma, with *r*_{0} = diam(*D*), to complete the proof of Corollary 2.2 and therefore of Theorem 2.1 as well.